We present a game-theoretic study of the selfish behavior of TCP users when they are allowed to use multiple concurrent TCP connections so as to maximize their goodputs or other utility functions. We refer to this as the TCP connection game. A central question we ask is whether there is a Nash Equilibrium in such a game, and if it exists, whether the network operates efficiently at such a Nash Equilibrium. Combined with the well known PFTK TCP Model [12], we study this question for three utility functions that differ in how they capture user behavior. The bad news is that the loss of efficiency or price of anarchy can be arbitrarily large if users have no resource limitations and are not socially responsible. The good news is that, if either of these two factors is considered, efficiency loss is bounded. This may partly explain why there will be no congestion collapse if many users use multiple connections.