When drilling plywood, it's a good idea to take precautions in order to prevent the drill bit from splintering the wood as it comes out the other side.
I just did a quick internets search for "drilling plywood" and all sorts of links came up. It seems I'm not the only one who should have known better.
The silly thing is that I noticed the splintering after drilling the first hole, but I just kept on drilling the rest of the holes anyway, without doing anything different. I guess I was in a hurry, since these chairs were supposed to be a surprise for my kids.
Of course, patching the splintered holes ended up adding a significant amount of time (days) to the overall project, as compared to taking five minutes to find a suitable second piece of sacrificial wood to place underneath the piece being drilled.