Wow...confusing answers, some say yes, some say no...Let's clear this up right now.
It is not the computer that gets the IP address, it is the network card (NIC) inside the computer thet gets its IP address from 1 of 2 ways. If you connect to a router, assigned by your ISP or if you bought a 3rd party router on your own you can have the router assign the IP address to your NIC automatically, this prevents all NICs that use this router from gettingthe same IP address. So if you have a computer or a laptop that has 2 NICS ( ethernet and wireless) your computer can have 2 IP addresses. You can also assign (if you know waht you are doing) the IP addresses yourself if you know the default IP address of your router. If you don't know how to do this, allow your router to do it. Now you mentioned that you may have two different routers as you have two different companies. Make sure that the routers from both companies have different IP addresses or you will get that nasty message that you have an IP address conflct which means that two computers trying to access the computer using the same IP address, even if they are on the different network, this will be caused because your routers IPs may conflict as well, They should have different IP addresses but if they dont they are easy to change if they use a 3rd party router such as belkin or Lynksys.