@Ian C. has provided an excellent answer to this question, but I can add a few thoughts. A few years back I was given a Huawei e160 USB from Vodafone mobile broadband here in the UK. The modem establishes a PPP connection over a 3G network, and since that network is shared with cell phones, Vodafone was aggressive about disconnecting idle devices after a very short time. They could always attribute these "dropped calls" to spotty network coverage.
@joshnh and @Martha may also be experiencing disconnections when their modems are idle for a short period of time. Or they may genuinely have spotty network coverage (sometimes signal strength drops for a few moments due to interference, then quickly recovers). In my experience when there was robust cellular coverage the PPP connection was good, and when the coverage was thin (in the countryside) or at peak times the connection became flakey.
Have either of you noticed that this problem happens more at certain times of day? That would point to your ISP shedding load.
Regardless, the solution is the same that @Ian C. suggested, namely to call your ISP.