Colette Berbesque and co-authors  decided to test the frequent famine assumption by looking at anthropological evidence of famine in a fairly sophisticated way. In particular, they controlled for the effects of habitat richness by only comparing hunter gatherers from warm climates (Effective Temperature > 13C) with agriculturalists, and by using a metric of habitat productivity (a linear combination of Net Primary Productivity and Effective Temperature) as a covariate. From their abstract, they found:
... if we control for habitat quality, hunter–gatherers actually had significantly less—Digging into the details though, we find that this is true for some measures of famine but not others. For instance, on the variables Occurrence, Severity, Persistence, Recurrence, and Contingency of famine warm climate hunter gatherers do better (that is, less famine). In terms of short term or seasonal famine however, there is no difference between warm climate hunter gatherers and agricultural peoples. And that's important, because it is seasonal famine that would lead to the evolution of thrifty genes. And there was plenty of time for such genes to arise in paleolithic people before the advent of agriculture, so the fact that agricultural peoples are no better at avoiding seasonal famine simply means the selection for these genes wouldn't go away after agriculture.
not more—famine than other subsistence modes.
Although Berbesque et al. paint their research as a critique of the "paleo diet" approach, it seems to me that it reinforces a key tenet: paleolithic peoples had it better than their agricultural neighbors. Hunter gatherers do significantly better on "ordinary nutritional conditions and endemic starvation", have no difference in seasonal famine, and less long term and unpredictable famine. Huh.
 J. Colette Berbesque, Frank W. Marlowe, Peter Shaw and Peter Thompson. 2014. Hunter−gatherers have less famine than agriculturalists. Biology Letters 10, 20130853.
*The Pima aren't necessarily a good example, as they were agricultural before becoming "modern western". It is interesting to me that the National Institute of Diabetes and Digestive and Kidney disease article I linked to focuses on the increase in fat in the Pima diet, rather than the exchange of whole grains for refined carbohydrate.