Does having two network cards increase internet speed?

Simply having two network cards does not increase you internet speed.

If those cards are connected to the Internet, they can increase your speed, your total speed being, in theory, the sum of the speeds of the connections attached to each of your two network cards. This setup however is problematic for home users, as requires a mechanism to balance the load on the two cards, can pose problems if you get to have multiple IP addresses etc.

Most likely, your answer is "no". However, if you wish to connect to multiple ISP's and increase your network throughput that way, look into "load balancing" and "multi homing".