Can I have two Internet connections -- one the VPN and one not?

The potential for confusion exists by your use of the term Internet. Multiple simultaneous network connections are possible. Each requires its own unique address. If you have a PC and it has two network cards, then they have two separate IP addresses to communicate by. (Unless you spent a lot extra for bridging cards and software!)

IF by the Internet you mean the Web, then the PC uses the faster connection by choice (metrics of transit times)....

Usually when some one says "VPN" they are referring to a connection from home or hotel to a company network. Most do not pass Web requests to the company connection. If you are using a DSL/cable modem and then running a VPN to the office, the Web requests should never go through the VPN. That would be an extra transit or two for the connection. If you are using dialup direct to the company network, you may be allowed Web access, but at dialup speed.