There are many possible issues at play, but let's start simple.
First, some terminology and assumptions. Your Internet connection is provided by your Internet Service Provider (ISP - usually cable, DSL, or fiber). This connection comes into your apartment and usually terminates at some kind of jack in your wall. You then buy a piece of equipment and plug it into that jack. That piece of equipment will be, for 99% of residential customers, a router that then allows your computers to connect to it to access the Internet connection. Those computers could be connected using wired connections or wireless connections. The computers and the router make up a local area network (LAN) which has access to the Internet via your ISP.
I will translate "make the internet run slower" to mean the computers in your apartment take longer to load web pages. If there is an issue, it could be either within your LAN or the ISP connection. These distinct options represent different reasons for the issue with potentially different solutions.
Computer networks are all about communication and, analogous to several people talking at once, the more computers that try to communicate at the same time, the less effective the communication is. So one possibility of reduced performance is that you have too many computers trying to use the connection. The router may be overwhelmed with connections and simply can't keep up (picture the I Love Lucy episode in the chocolate factory). When this happens your computer has to keep sending the same data until the router is able to handle it. Another possibility is that the router is just fine, but the connection to the ISP is full. The router is probably trying to be fair and gives each user a little slice of that ISP connection so from your perspective everything seems to get slow and unresponsive.
The deluge of communication could be legitimate traffic or it could be due to problems on the computers you're using. If one has a malicious program running it may be sending out garbage and overwhelming the router or ISP connection. If one of the computers has a hardware problem or a driver problem it may be sending out garbage in the same way. Updating the drivers and operating system on the problem machine (Mac or PC) should help. Updating the firmware on the router may also help.
But let's suppose that the router is plenty robust and the ISP connection is sufficient to handle the traffic and all the hardware is working as intended. In this case it's still possible that including a new machine could result in poorer performance for everyone. I am assuming all of your machines are connected wirelessly. Suppose you have a fairly new router that supports the faster 802.11n protocol and that all the non-Mac machines you use also support the 802.11n protocol. When this is the case everything is operating great at the faster speeds allowed by the 802.11n protocol. Now suppose that the Mac connects, but the Mac doesn't support 802.11n, so it connects using the slower 802.11g protocol. This can actually cause everyone's connection to slow down to the slower speeds, which is lame. SmallNetBuilder has a great article showing this happen.
Now all that being said, is there something specific about Macs that could, in general, cause network problems? Well, I don't have any Apple products so I don't have any personal experience with the matter, but, according to a Google search, your roommate isn't the only one who has experienced/believes the phenomenon. In fact the Apple support forums are filled with threads of people having their Mac crash their network. A search for "Mac crashing router" results in 418,000 hits in the past 12 months. "Windows crashing router" results in 1,130,000 hits in the last 12 months, but since the Windows install base is approximately 10 times larger this shows a disproportionately high number of hits for the Mac search.
So there very well may be some truth to your roommate's position, but it certainly shouldn't be true for all Macs.
-Curious Physics Minor