Well, after only 16 months, my phone has broken. Right in the middle of my using it, the phone went dark. After waiting for a couple minutes for it to come back to life, I decided to pull the battery and restart. Unfortunately it didn't restart. The boot screen appeared briefly, but went off again and the phone started vibrating every 2 seconds.
A bit of googling suggests that the power switch has gone bad/stuck and shorted out or something, and that it's a relatively common problem. This thread over at the xda developers forum suggests a possible fix or work around to try.
Since the phone is a few months out of warranty, I might give it a try. The alternative would be to send it in to Samsung to see if an out of warranty repair could be done. Since it is a known issue, maybe they'd even be able to do it as a warranty repair. Will have to get in touch with them to see.
I don't feel like I've had this phone long enough to warrant getting a new one, although the S4 looks pretty nice. I don't think we could squeeze it into the budget right now though.