Can I make potatoes sweet by cooking them?

What you intended won't work. I have also heard generic statements that cooking in principle breaks down complex carbohydrates to simple ones, but this doesn't mean that all complex carbohydrates will break down, nor that any cooking method will do it. And when we come to your specific example, it is a no: starch doesn't break down into sugars at potato-cooking temperatures. Even if you were to heat starch more, it doesn't become especially sweet - you can observe it when making dark roux.

There are other ways to break the starch down, and the other answers point out the main ones, cold temperature and enzymes. But not cooking.

A raw potato is 79% water, 17% carbohydrates (of which 88% is starch) and 2% protein. So your question is essentially: can starch be turned into sugar?

In theory the answer is clearly 'yes': this is what plants do when they break down the starch created as the final product of photosynthesis, and what happens in the human digestive system. Starch degradation is a complicated process involving several different amylase enzymes.

In practice, I think the answer is 'no'. In a raw potato, starch is present in many small granules. When cooked, the starch gelatinizes: the granules burst and water is absorbed. This is a irreversible chemical change that obviously has a big effect on taste and texture. But it does not produce significant amounts of sugars.

A closer match to the starch degradation process in plants occurs in brewing. When beer is brewed from barley by heating a 'mash': the natural amylases present in the grain convert starch to sugars; these are then fermented by yeast. The exceptionally sweet-toothed might try to mimic this process with potatoes, adding something like wheat, barley or sweet potato that has natural amylase. You could then boil the potato 'proto-beer' to extract the sugar.

The way to make potatoes sweet is actually to not cook them right away -- you need to chill them, to give the time for starches to convert to sugars. (it's called "Cold Induced Sweetening"), which is discussed in Serious Eat's take on Austrian Potato Salad)

If you know someone who grows potatoes, they can just leave them in the ground to dig up in the early spring. For the rest of us, it means leaving a bag of potatoes in the back of the fridge for a few weeks.