Is it possible to study abroad as a PhD student? I hope to enter a PhD program in economics in the next few years, ideally at a top US school (Berkeley, Stanford, MIT, Harvard, etc.) Are there usually opportunities to study abroad during such a program? For example, if I attended Berkeley intending to focus on macroeconomics, would it be possible to study abroad with some of the macro theorists at Cambridge or U of Edinburgh?

This would not be an opportunity to "experience the local culture," which many undergraduate study abroad experiences focus on, but more a chance to gain multiple perspectives from academics in different countries. In my opinion the rigors of a top PhD program rule out the former.
I don't think what you're describing is all that common, and I've never heard of any sort of systematic program, but it does happen.  I ended up spending two months abroad during grad school to participate in a once off trimester program in my area of specialty.  There weren't any standard arrangements in place, though: my home university (very generously) moved things around so I could do it.  I think I've heard of other cases, all similarly involving unique circumstances like faculty moving around or particular personal connections.