The best answer probably ranges somewhere between 'yes' and 'maybe'.
T-Mobile Digits is one route. That service allows calls/texts to be pushed to multiple SIMs with unique IMEI numbers but all SIMs on said plan end up using the same phone number. All devices can be online simultaneously.
Another route is cloning a SIM (in the legal sense). By duplicating a working SIM you can have two identical SIMs in your two phones, however, only one device can be online at a time. One device must be fully powered down before the other can be powered up and registered on the network. If both SIMs ever hit the server at the same time, chances are high that you'll lose service due to either a network fault or by triggering a SIM clone error in the illegal sense. Since each phone has a unique ESN/MSN, you're better off with a carrier like T-Mobile purely because they don't throw a fit with provisioning. AT&T can work if they play well with both devices and will provision them for full service. VZW might work as well but again both devices would need provisioned individually before cloning a SIM for use in each device. The same caveat applies about one device going offline before the other goes online.
With that said, if you aren't already on TMO then I wouldn't switch carriers just for this.
There's a much longer answer to provide here but hopefully this gets you the basics.
Edit: A third option might be to use one phone for calls/texts and the other phone for everything else; web, apps, etc. You would obviously only get messages on one of two phones but maybe this would be a solution.
A fourth option might be something along the lines of Google Voice, although you might have to port your number over to GV for that to work on two phones. It's been a while since I've last looked into that. Google might be of some help here.