In the long run, NO! The base oils are all compatible, it's the additive packages that are not complementary and indeed some additives from one manufacturer can interupt the function of another - for example some "thickeners" (aka viscosity enhancers) used in petroleum motors oils can interact together plus with other agents and make the sewage sludge mess mentioned in other posts. Another good argument for synthetic oil, which doesn't need these additive just to pass some rather low hurdle of a standard test.

OK I am a biased Amsoil dealer, but then again, you become biased when you don't even need to look at your flame trap because you know it won't clog with Amsoil!
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http://www.fidalgo.net/~brook4/oilslubesfilters.html