Technically yes, it's feasible to do so, and should be functionally okay. You're not going to get the full speed of plain old cat 5 but I would imagine you'll still reach 100 on a moderate run with cat5e, shielded would even be better. just ensure that you're wiring the right pairs. (Green and orange for ethernet data)
Downsides I can think of is interference when the phones are on use or ringing, phones use quite a bit of voltage, I think the phones would be clear though, the internet might have some troubles. I'd be interested in hearing the result!