Starting in 2006 many airlines have cracked down on interlining on separate tickets. Most will not interline except within the same alliance. All will interline on single tickets. And prior to 2005 it was no problem to interline with separate tickets.

So it all depends on the 2 airlines involved. Some have agreements and will do it with separate tickets, some won't. And many of these agreements are not symmetrical (so you can sometimes interline in one direction, but not on the return trip, for example). The airlines do this to get you to buy a single ticket. They have figured out that people are buying separate tickets to save money, and are using interlining as one of the justifications for paying more for a single ticket. Your alternative if you must check your luggage through is to ship it as air freight (costs for this vary, but is typically a few hundred dollars).