What would cause a handling problem after installing new tires. The car has had a front and rear alignment and it didn't help Car was fine before the tires were installed?

This could be caused by low tire pressure, very weak springs, or worn out struts or shock absorbers. I would check these things first.

Describe the problem in more detail. You can feel the back of the car moving back and forth at a speed of 65+ on the highway. The alignment folks think it must be the tires. They took the car back and looked over everything again and saw nothing mechanically wrong with the car. They suggested the tires may be to be "broke in" or get about 500 miles on them before they handle right again. I've never heard of such a thing. Is this normal or is it the brand of tires that were put on. The people who did the alignment didn't put on the tires.