Did ancient Romans anticipate the fall of Rome?

Who are these Romans you are talking about ?

When we talk about final years of (Western) Roman Empire, first thing we need to know is that population from that period greatly differed from Cives Romani during the Republic, or even their Latini neighbors. In fact, most of late Roman emperors would not be considered as Romans just few hundred years before. For example, Constantine The Great had Greek mother, he was not born in Italia and even origins of his father were suspicious.

During its final years, Roman Empire was territorially very large, but as multi-ethnic state very weak with little cohesion by different parts except military might and in some cases inertia of population. And even that military might was failing - Roman army was no longer citizen army fighting due to sense of duty and honor (like for example in Punic Wars). Instead, it was purely mercenary organization, sometimes falling so low to bribe and hire barbarian tribes instead of training their own soldiers.

Most of the population in bordering provinces like Britania or Germania did not have Roman citizenship, they were simply subjects of Rome, more or less unwilling. When central rule started to collapse and new barbarian warlords appeared, they simply switched their allegiance. Of course, in some cases new rulers were in fact former Roman officials which retained some parts of Roman civilization, which made transition from Roman to non-Roman rule smoother and easier.

On the other hand, population in central provinces (especially Italia ) had gotten used to political instability, change or regimes, and occasional military campaigns and looting by various barbarian mercenaries. It was a gradual process, but, as mentioned before, by the time it happened most of the population in Italia did not have Roman or even Latin ethnic background, and they were not particularly motivated to defend failing state. They were simply waiting to see who would end up strongest in the end.

Finally, we have provinces of Eastern Empire which gradually become Byzantine Empire. Although nominally Roman till the end, this part of empire had predominant Greek element, with Greek language (with associated rich culture and literature) simply replacing Latin as spoken and official language. With Greek ethnicity in the core of this new state, and Constantinople as new capitol, Rome was gradually "out of sight, out of mind". Byzantine Empire did occasionally hold parts of Italia, and even Rome itself, but real center was elsewhere - subjects of Byzantine Empire didn't share much with ancient Rome except the name.

Fall of the Roman Empire and Roman civilization was in fact caused by lack of people willing to defend it . In late 4th century when terminal crisis started, there were no more ancient Romans willing to pro patria mori. Romans died before the Rome, which is a pattern commonly recurring at the end of each great empire.

Scipio Africanus the Younger is reputed to have foreseen the fall of Rome at the very moment that he oversaw the destruction of Carthage. There are several versions of the story here

A glorious moment, Polybius; but I have a dread foreboding that some day the same doom will be pronounced on my own country

I very much doubt you can find evidence of anybody expecting the fall of the (Western) Roman Empire. The Eastern Roman empire did survive and had a number of historians who documented the events which happened between 375 and, say, 600. Two of most prominent ones are Zosimus and Procopius. As you can imagine, they were rather horrified by what was happening. Here is Procopius (in Secret Hisory):

Italy, which is not less than thrice as large as Libya, was everywhere desolated of men, even worse than the other country; and from this the count of those who perished there may be imagined. The reason for what happened in Italy I have already made plain. All of his crimes in Libya were repeated here; sending his auditors to Italy, he soon upset and ruined everything.

The rule of the Goths, before this war, had extended from the land of the Gauls to the boundaries of Dacia, where the city of Sirmium is. The Germans held Cisalpine Gaul and most of the land of the Venetians, when the Roman army arrived in Italy. Sirmium and the neighboring country was in the hands of the Gepidae. All of these he utterly depopulated. For those who did not die in battle perished of disease and famine, which as usual followed in the train of war. Illyria and all of Thrace, that is, from the Ionian Gulf to the suburbs of Constantinople, including Greece and the Chersonese, were overrun by the Huns, Slavs and Antes, almost every year, from the time when Justinian took over the Roman Empire; and intolerable things they did to the inhabitants. For in each of these incursions, I should say, more than two hundred thousand Romans were slain or enslaved, so that all this country became a desert like that of Scythia.

Such were the results of the wars in Libya and in Europe. Meanwhile the Saracens were continuously making inroads on the Romans of the East, from the land of Egypt to the boundaries of Persia; and so completely did their work, that in all this country few were left, and it will never be possible, I fear, to find out how many thus perished. Also the Persians under Chosroes three times invaded the rest of this Roman territory, sacked the cities, and either killing or carrying away the men they captured in the cities and country, emptied the land of inhabitants every time they invaded it. From the time when they invaded Colchis, ruin has befallen themselves and the Lazi and the Romans.

Here is Procopius in History of Wars, III:

Afterwards Gizeric devised the following scheme. He tore down the walls of all the cities in Libya except Carthage, so that neither the Libyans themselves, espousing the cause of the Romans, might have a strong base from which to begin a rebellion, nor those sent by the emperor have any ground for hoping to capture a city and by establishing a garrison in it to make trouble for the Vandals. Now at that time it seemed that he had counselled well and had ensured prosperity for the Vandals in the safest possible manner; but in later times when these cities, being without walls, were captured by Belisarius all the more easily and with less exertion, Gizeric was then condemned to suffer much ridicule, and that which for the time he considered wise counsel turned out for him to be folly. For as fortunes change, men are always accustomed to change with them their judgments regarding what has been planned in the past. And among the Libyans all who happened to be men of note and conspicuous for their wealth he handed over as slaves, together with their estates and all their money, to his sons Honoric and Genzon. For Theodorus, the youngest son, had died already, being altogether without offspring, either male or female. And he robbed the rest of the Libyans of their estates, which were both very numerous and excellent, and distributed them among the nation of the Vandals, and as a result of this these lands have been called "Vandals' estates" up to the present time. And it fell to the lot of those who had formerly possessed these lands to be in extreme poverty and to be at the same time free men; and they had the privilege of going away wheresoever they wished. And Gizeric commanded that all the lands which he had given over to his sons and to the other Vandals should not be subject to any kind of taxation. But as much of the land as did not seem to him good he allowed to remain in the hands of the former owners, but assessed so large a sum to be paid on this land for taxes to the government that nothing whatever remained to those who retained their farms. And many of them were constantly being sent into exile or killed. For charges were brought against them of many sorts, and heavy ones too; but one charge seemed to be the greatest of all, that a man, having money of his own, was hiding it. Thus the Libyans were visited with every form of misfortune.

And here is Zosimus (in Book V of Hisoria Nova) writing about Alaric's invasion (about 100 years after the fact):

Alaric on this marched out of Thrace into Macedon and Thessaly, committing the greatest devastations on his way. Upon approaching Thermopylae, he privately sent messengers to Antiochus the proconsul, and to Gerontius the governor of the garrison at Thermopylae, to inform them of his approach. This news was no sooner communicated to Gerontius than he and the garrison retired and left the Barbarians a free passage into Greece. Upon arriving there, they immediately began to pillage the country and to sack all the towns, killing all the men, both young and old, and carrying off the women and children, together with the money. In this incursion, all Boeotia, and whatever countries of Greece the Barbarians passed through after their entrance at Thermopylae, were so ravaged, that the traces are visible to the present day.