networking - Can you run DSL and T1 through a single switch for two different purposes?

I'm not quite sure this works, but my co-worker says it will. Details first: We have a small office of less than 10 PCs, and we use VOIP for our business phone line. All PCs require internet access. Now then, we used to run everything through the DSL line, but it would slow down a lot, so we recently had a separate T1 line installed, with 10 dedicated IPs. Our office is equipped with one switch, and the idea came up to separate PC traffic from VOIP traffic.

My co-worker suggested that we connect the T1 line to our switch and use the dedicated IPs for each PC. Then we connect the DSL line to a router and connect that router to the SAME switch, so that it can use DHCP for the VOIP phones. The end result would be that all office PCs would be using the T1, and all office VOIP phones would be using the DSl line. One small catch: There is only one network wall socket in each office (all connected to the switch), and the current office setup has each PC connected to the VOIP phone, and then the VOIP phone connected to the wall socket.

Is this possible? Two different internet connections (DSL and T1) through the same switch, two different machines (PC and VOIP phone) essentially sharing one connection to the switch (since the VOIP phone sits inline with the PC), and having the PC only use the T1 and the VOIP phone only using DSL?