You certainly can have 2 wireless networks going.
One option is to have a WAP - Wirless Access Point, essentially a bridge between the wired and wireless networks, plug it into your existing router’s wired network ports. Silly thing is, this is simpler than a router, but less common, so hence a lot more expensive usually.
Another is to cascade routers; this works fine too. Your new router WAN port connects to a network port on the old router; instead of PPPoE or whatever, just use “acquire a address automatically” (DHCP and plain ethernet) for the router’s WAN port, assuming the old router has DHCP turned on. Your connection may be a little slower, but it will work.
The third alternative is to turn your new router into a WAP - this is simple. Turn off DHCP on the new router, and connect one of its internal network ports to a network port on the old router. No connection to the new WAN port.
Because then the new router is set to not do DHCP, it does not hand out addresses and does not tell anyone it is the default gateway, so nobody tries to send the packets to its address to be forwarded; they get their default gateway with DHCP address from/as the old router. It simply acts as a switch extending your network. If you want to manage it, you will have to set its internal address to something compatible with the old router. (I.e. old router is 192.168.0.1 hands out 192.168.0.100 to 150; set new router to 192.168.0.2, and you can login and manage it from the old/combined network). The wireless portion is just bridged into the network, but with its own SSID and security settings.
Of course, to connect the two routers you may need a crossover ethernet cable (1,2 goes to 3,6 and 3,6 goes to 1,2). And while you’re at it, it’s probably a good idea to separate the two routers as much as possible so to lessen interference.
Some fancier home routers can even be programmed as a wireless bridge and WAP to make a wireless extension to an existing network; but that depends on the router and gets a bit more technical.
So your roommate doesn’t know as much as he thinks he does; or he’s trying to BS you; or more likely, both. If he’s trying to stop you from having an unsecured network, he’s probably got a point. (If you are using WEP-128 and need that 26-character hex key, try putting a $ (dollar sign) in front of it when you enter it into the iPhone - worked for me in 2 networks so far, hex all in caps). If he’s just turning off he SSID, first it’s not a great security measure to do so, second why bother? If he’s as good as he says and does this for a living, he should be able to get your iPhone working with your wireless and security.