1999 Toyota Camry rattling/boucing rear

That huge sentence doesn't make sense. Please put some punctuation in it so I can figure out which way to read it. Are you saying something is making a rumbling sound like a drum? Did this noise start after parts were recently replaced or was it there before? What was done to the brake system? What parts exactly were replaced?

Have you noticed if the noise stops when you're applying the brakes lightly or when you're turning a little?

Dandy. If you feel a brake pedal pulsation or a shimmy or wobble in the rear when the brakes are applied, there is an issue with the brake drums. The drum is separate from the hub / bearing assembly. They're held together by the wheel studs, and it's not uncommon to accidentally assemble the parts off-center. That, and not tightening the wheel nuts to the correct "torque" or tightness can lead to them warping. A drum can arrive warped from the auto parts store too but usually not bad enough to be noticed.

For the noise, there is a tool you might be able to borrow or rent from an auto parts store that borrows them called the "Chassis Ear". It is a set of six microphones, a switch box, and headphones. You clip the microphones to suspect points, then drive around while listening with the headphones. You can move the microphones around to zero in on the source of the noise. Be aware that many mechanics have never seen or even heard of this tool. Suspension and alignment mechanics use it to find rattles, squeaks, and other noises.

There's too many parts to list for the rear suspension and brake systems, but none of them have an unusually-high failure rate on your car model. Your mechanic should have noticed if the noise was still there on the test drive after the alignment. The next thing would be to have the steering and suspension systems inspected at a tire and alignment shop for a second opinion. They're experts at finding the causes of noises and vibrations, and they will "read" the tire wear patterns for clues to the problem.