Say I have two passports from two countries, both of which will allow me to stay in country X for 60 days without requiring a visa.
Can I enter with passport A, stay 60 days, then leave and immediately re-enter with passport B to stay for another 60 days?
You actually can do this, I've met people that use this strategy. Here's the catch, generally you aren't in their system, since passports are very complex, and from what I've heard the EU isn't very good with keeping track of all of it. BUT if you get caught you can get blacklisted from visiting again.
Along with a fine, and possibly seeing a judge. This isn't legal advice, but really better off not doing this. You may get away with it for a bit, but the problem you'll run into is when you get blacklisted on BOTH passports for 5 years or whatever they decide.
I've seen other travelers get penalties ranging for 1 year to 10 years of no entry, in various countries due to messing around with visa laws. It's ill advised in any country, not just EU zone travel. The penalty obviously depending on the issue.
There are many legal and legitimate ways of extending your visa past the 3 months in, 3 months out rule that they impose. You can sometimes take as little as 1 class and qualify for a longer term visa in some EU countries. Other places will give you residency for starting a business. There are options. And using the dual passport "trick" could end up with you being in big trouble at worst, and running the risk of being blacklisted at best.
TLDR: Not advised at all.
Generally, switching passports does not help. (It could in fact hurt if it is noticed, because an officer might take exception to the apparent intent to deceive.)
Most countries' laws are expressed in terms of persons. The law typically says that a person who is entitled to a visa-free stay is permitted to stay for some certain length of time. The fact that the person has two ways of qualifying for visa-free stays (i.e., two visa-exempt nationalities) does not change the fact that the person is entitled to stay for that certain length of time. An example of this is the Schengen area, where third-country nationals must abide by the 90/180 rule.
Similarly, some countries allow people to renew their period of stay by leaving and immediately re-entering the country. Doing this with a different passport is no better than doing it with the same passport. An example of this is the United States (immediate re-entry may cause suspicion of not being a genuine visitor, but it is not explicitly forbidden by law or regulation).
There will be exceptions to this if the two nationalities have different rights or privileges in the country in question; in that case, one might be able to extend a stay if they started out with the less privileged passport and subsequently showed the more privileged passport. An extreme case of this would be an EU country where one of the traveler's two countries of citizenship is also an EU country. But that's not a very satisfying example and doesn't seem to be the sort of situation you're asking about.
There are some questions here that consider this for specific cases:
