Yes, you should be able to assign the same SSID to all of your wireless access points (WAPs) and clients should be able to roam between them.
HOWEVER, when doing this kind of setup, the clients will try their best to stay on the same AP. If and when they roam to the next one, the client will actually disconnect momentarily and re-lease it's IP from DHCP. So if you were streaming a movie, then roam to another WAP, the movie will stop playing.
In a true multi-WAP setup (where the WAPs are all acting in concert, not merely configured the same way), there will be no disconnection when roaming. This is due to the way the system will broadcast itself and because, usually, these enterprise systems will tunnel traffic from clients back to the firewall/controller.
You should also be careful with co-channel interference from WAPs that are too close together. As you know, channels 1-6-11 are non-overlapping channels... so if there are two APs on the same channel that can "hear" each other, you will take a performance/reliability hit. Especially on a network such as yours where all the WAPs are not part of a fully-integrated system.
Regarding your DHCP environment, I don't feel that I interpreted your setup 100% correctly, but there should be only one DHCP server in the same Layer-3 address space. I understand that you have two modems, so I ASSUME you DO have multiple DHCP sources. However, since you seem to be running independent WAPs, this shouldn't be a problem... as long as they can't see each other. Now, your clients might get confused as they roam from DHCP domain to DHCP domain... And of course, users will only ever have access to one of the internet connections (modems) at a time.
A better solution would be to go with a fully-integrated solution with mesh WAPs that can all talk to each other and act as "over-the-air" routers. This way, you could have one DHCP server for the entire setup, and traffic can be automatically routed to the best internet connection (modem). These kinds of networks are also self-healing, which means that if a WAP goes down, the remaining WAPs can route around it. This redundant, meshed solution will also allow users to SEAMLESSLY roam from WAP to WAP without disconnects.
A fully-integrated system such as I'm recommending would also allow you much more control over how things work, who can get connected, configure multiple wireless SSIDs on the same WAPs, configure VLANs, Layer-2/3 ACLs, captive portal, more secure (and easier!) authentication mechanisms, and much more.
Full disclosure: My company (CWB) is your Ruckus Wireless solutions provider in the Middle Tennessee area.
I was afraid that would be your answer. I'm dealing with very non-technical users who I have limited-to-no communication with, so my preference would be a single SSID, even if it means a few dropped packets when they hop between gateways. The ISP offers a plan that has 2x the speed of what either modem is getting, so probably I could just use a single modem and the three routers. I don't want to run any cables, but we don't always get what we want. – Alec Munro – 2011-09-21T18:30:09.520
Unfortunately, they won't just hop between gateways. Just name the SSID's "MyApartment1" and "MyApartment2". The connection software on people's machines will do well enough helping users pick the one with the stronger signal. – Joel Coehoorn – 2011-09-21T18:31:49.727
This looks like a good candidate to trunk the modems together: http://www.cisco.com/en/US/products/ps9924/index.html Runs about $400, but it should be a solid piece of a equipment.
– Joel Coehoorn – 2011-09-21T18:33:13.810Joel is right. Having the names the same caused serious problems. :( – Alec Munro – 2011-09-22T17:20:23.757