From a New York Times review (Don't Blame Columbus for All the Indians' Ills) of the book The Backbone of History: Health and Nutrition in the Western Hemisphere:

What had not been clearly recognized until now, though, is that the general health of Native Americans had apparently been deteriorating for centuries before 1492. ... More than 12,500 skeletons from 65 sites in North and South America -- slightly more than half of them from pre-Columbians -- were analyzed for evidence of infections, malnutrition and other health problems in various social and geographical settings. ... The surprise, Dr. Armelagos said, was not the evidence of many infectious diseases, but that the pre-Columbians were not better nourished and in general healthier. ...

The more mobile, less densely settled populations were usually the healthiest pre-Columbians. They were taller and had fewer signs of infectious lesions in their bones than residents of large settlements. Their diet was sufficiently rich and varied, the researchers said, for them to largely avoid the symptoms of childhood deprivation, like stunting and anemia. Even so, in the simplest hunter-gatherer societies, few people survived past age 50. In the healthiest cultures in the 1,000 years before Columbus, a life span of no more than 35 years might be usual. ...

The researchers found one exception to the rule that the healthiest sites for Native Americans were the oldest sites. Equestrian nomads of the Great Plains of North America in the 19th century seemed to enjoy excellent health, near the top of the index. They were not fenced in to farms or cities.

Hunter-Gathers

Absolutely all hunter-gatherers live "hand-to-mouth", malnutrition is common, and starvation is not just a "risk" - it is a permanent threat.

This should be obvious because they cannot effectively store excess food and thus are subject to the standard predator–prey model:

plenty of food --> 
population expansion -->
depletion of food sources -->
population contraction -->
plenty of food

native societies were relatively very stable

Medieval inner city looks like a modern college campus "safe zone" compared to the level of violence in "native societies".

People could could call on a wide range of resources

This is an exaggeration: a tribe hunting deer will probably have little to fall back on, especially in winter.

social networks

A tribe is a single social unit. They hunt together, they eat together, they starve together. Remember, they cannot store food! Yes, some have a better teepee or moccasins, but not food.

A neighboring tribe might fare better (unlikely, but possibly), but the distance kills cooperation. They are not likely to be bosom buddies (they attack each other to kidnap women all the time), and transporting food is very hard.

Agricultural Societies

These fared better, but not by much.

They did not have high-yield crops and draught animals (and thus the wheel).

The first meant that they still lived hand-to-mouth (even though sightly better than hunter-gatherers because grain store better than meat) and the second meant that a local crop failure (due to, e.g., a drought) could not be mitigated by import.

See Guns, Germs, and Steel for details.

Food Storage

Effective food storage is a relatively recent invention. Pre-industrial societies did not produce much excess food for storage and could not store if very well.

E.g., GurvenKaplan2007 mention

the case of an Nunamiut Eskimo group that perished in its entirety, having been snowed in without sufficient food supplies to survive through the winter.

PS. I am not saying that if a man fails to catch his daily quota of fish/fowl/venison, then his family will starve tomorrow. They can survive on the combination of yesterday's catch and the wife's gathering. Death from starvation was probably not an annual event. However, hunger was.

PPS. Further reading:

• Hunger and Work in a Savage Tribe