So it turns out it was not a dog bowl Bill saw sticking out of the ground. No, it was actually a frying pan missing the handle. (Which I later found in the middle of the yard this afternoon). More bones have yet to be sighted, so my current theory is that someone killed someone with a frying pan, and buried the evidence in our backyard.
Seriously, though, I think that someone just used the back corner of our yard as a trash pile in the past. I found a rope and part of a cottage cheese container today, too. I showed Michelle the two bones today, and she doesn't know what to think either... So we'll see if anything more turns up.