On what basis do Christians sometimes claim to know what God's intentions are?

I may be wrong, but this doesn't so much sound like a question about Christianity as it is about human nature. I can think of four scenarios where I've heard people speak as if they know the mind of God.

1. When we are asked a question, we feel compelled to give an answer. If we don't know the answer for sure, we tend to give an answer that we think SOUNDS good. (Hey, I'm doing it now!) We don't like to feel stupid, and when we can't answer a question or defend our beliefs with good, solid facts, we make up the best answer we can.

   In the example you pointed out, the answer being given is obviously along these lines. If I were a betting man, I'd bet that this was a well-intentioned creationist that was confronted with a question he or she did not know the answer to. Instead of admitting ignorance, he or she gave the answer that made the most sense to him/her.

2. In the comment to the question, Kristof Claes gave another very good example about the natural disaster as a punishment for sin. In those situations, I agree with you 100%. It is presumptuous to assume to know the mind of God. We, unlike God, are finite in our intelligence and understanding.

3. When going through suffering or trials ourselves, we tend to try to figure out why God is allowing this to happen to us. I have two relatives, for example, who prayed for patience, and ended up with broken legs shortly thereafter. (True story.) They assumed that the broken legs were an answer to prayer (although not in the way they had envisioned when praying.)

4. Finally, along the same lines, I've also had friends that were going through trials and assumed that God is testing them, or wants them to learn something, etc. In these cases, they tend to say "I don't know why God is allowing this, but I assume it's because... " or "God must want me to learn something from this." these types of statements are based on what the Bible teaches us about why God allows us to go through trials. (I'll provide verses if necessary, but I don't think they are in this case.) They are saying that they know that god has a reason (assuming to know something about the mind of God), but not exactly what, which puts these in a different category from what I believe you were getting at.

This is what Luther calls the theologia gloriae.¹

The most obvious way to the cognition of God is to follow the footsteps of creation. The creation speaks a powerful language.² Scholasticism has understood the natural cognition as a way of salvation.³

Scholastics think of God and man from the same viewpoint of natural causality and connect the cognition of God per ea, quae facta sunt⁴ with a particular ethical view of man and his works.⁵

This connection builds the base, upon which Luther critisizes the theologia gloriae: That person does not deserve to be called a theologian who looks upon the »invisible« things of God as though they were clearly »perceptible in those things which have actually happened«⁶

For him, real cognition does only happen by cognition of Cross of Christ: theologia crucis.

And finally a biblical quote (found this here):

Matthew 24:36 (NLT)
Mark 13:32 (NLT)

 ³⁶ “However, no one knows the day or hour when these things will happen, not even the angels in heaven or the Son himself. Only the Father knows.

^{1: See Heidelberg Disputation, Thesis 19
2: Walther von Loewenich, ''Luthers theologia crucis'', p.19
3: About the question what scholastics say about Rom 1:20, see Pierre Bühler, ''Kreuz und Eschatologie'', p.106
4: literal translation: ''by the things that have been made''
5: Theodor Dieter, ''Der junge Luther und Aristoteles'', p.109
6: WA 5,388,5f.: Non ille digne Theologus dicitur, qui invisibilia Dei per ea, quae facta sunt intellecta conspicit}