This is sometimes a debate, whether or not an electrical device like an amp breaks in like an electromechanical device, like a speaker. So I'm going to put it out there: I think they do.
There's a lot of voodoo around. There are people who think cables break in. Um...no. There are people who believe that electrical cords break in. Uh...not really. But amps...amps are different for some reason. They pass a lot of current. They get hot, stuff happens to the transformers and the parts, and I think you can hear it.
My HX/DA has really opened up in the month or so since I have had it. I really noticed it doing a session yesterday. It's hard to explain, but it just sounds sweeter, feels a little looser, and is altogether more wonderful than an amp has a right to be.
This is just my opinion, of course. What do you think about whether or not amps break in?
Note: I'm referring to tube amps, primarily, not modelers, and not solid state.