Out of 8 K-1's, all of mine have Box 16 checked, but NONE of them have ever sent a K-3, and only one has some foreign taxes paid.
In TurboTax, it asks if Box 16 is checked, but then when you're asked later whether you actually received a K-3, it states that if you don't have foreign income or tax associated with the K-1, you can "uncheck" it, or say "No, Box 16 is not checked" (even though it is).
