Eating improperly handled meats and contaminated vegetables can cause digestive problems or worse. My intuition is that prior to the widespread availability of refrigeration, and especially prior to mass knowledge of germ theory, it would have been very difficult to avoid eating tainted items, and thus getting sick. On the other hand, constant exposure to foodborne pathogens would have presumably caused people to develop at least some amount of immunity.
Do we have any reliable information about that? To make the question more specific, as an average adult, roughly how many times a month could I expect to have diarrhea or an upset stomach in your time period of interest?
I can only really speak with regards to Britain in the early modern period (18th and 19th Century) based on a few sources that I have looked at. It would be hard to say as to how often someone would get sick, as obviously it would vary person to person and there would certainly not be any statistical data to outline this. What I can however display is that it was a concern, and that people were loathed to buy meat or produce that was past its best and spoiled. Books were commonly produced with techniques as to how to buy the freshest ingredients/meat/products available.
This ranged from the smell and appearance of meats, and even ways to test the produce. It is important to remember that food shopping in this period was vastly different to today, with even poorer families preferring to buy cheap food at inns and taverns rather than buy and cook produce themselves. For those that did go to market daily and purchase food (the women usually serving this important domestic role), there was a genre of instructive literature purely to assist in this.
I'll include an extract from the source detailed below to give a kind of illustrative idea about the practices necessary in order to make a "Compleat Market Woman". Hopefully it will also serve to highlight that they were aware of when meat and produce was spoiled, and for a large part, I would speculatively suggest that illness was avoided to an extent greater than we would imagine through using primitive and rudimentary knowledge.
"EGGS
To know the Goodness of an Egg, clap your Tongue to the great End; if you find it has any Warmth, depend upon it 'tis new, but on the other Hand, 'tis bad, if it be quite cold.
Another Way
To discover whether an Egg be good or bad, put it into a Pan of cold Water; if it falls directly to the Bottom, 'tis fresh; if it swim at the Top, depend on it 'tis rotten.
How to preserve them for Months, if good when bought
Put them into fine Wood Ashes, with their small End downwards, and turn them End-ways once at least every Week."
"The Young Woman's Companion" by Mary Johnson (1753)
Let me know if you need any more advice from Mrs Johnson... Hopefully that gives you a kind of idea of the level of knowledge/precautions that were taken in order to try and combat illness and waste before refrigeration.