We caught Whitey disassembling one of the fountains and figured out how he does it. He reaches over the top of the cover, hooks his claws over the edge, and pulls. This keeps his paws from getting wet in case he forgets to unplug the fountain first.
Once that's done, it's easy for Billy to grab the filter. We haven't figured out how they manage to pull the reservoir out -- essentially a big plastic bottle -- but it was in the middle of the dining room floor this morning.
The boys were in fine form in other ways this morning, too. There are two entrances to our front hall, one from the dining room and one from the living room. Both boys went charging toward the hallway cat tower, one through each entrance. They both leaped at the same time... and collided in mid-air. No harm done, and Billy even recovered quickly enough to seize the high ground in the tower.