I had tried to assemble the components for a standalone atmega168 on Radioshack perf boards (two different kinds, the ones with copper pads around each hole, and the kind set up like a breadboard), and the traces kept separating from the board and breaking connections.
Sounds about right. Don't but anything from Radio Shack, their stuff is all garbage. That will not happen with a decent board (which those are not) unless you really overheat it with your iron. I hope for your sake you're not using a cheap radio shack style soldering iron. About all those are good for is turning people off electronics for life.
Remember that the bond between that pad and the board is all that mechanically holds your component in place on the board. It has to be strong.
I don't see how the RBBB will help in this case, you'll end up making almost as many connections between this board and the other as you would to put the ATMEGA right on the other board.