GFCIs tripping on their own
Shared neutrals could cause this.  If the two circuits are both in a box together somewhere, and the neutrals from the two circuits are tied together, this could cause this problem.

It's common practice for DIYers (and some "professionals") to simply connect all the neutrals in a box together. If there are two circuits present in the box, simply connecting all the neutrals together is not proper. In the past, this normally didn't cause any noticeable problems. However, now that GFCI and AFCI protection is common, this bad wiring practice becomes more apparent.