Quad Gigabit Ethernet Cards....advice needed.

I think a little backstory is needed here.

What is this setup used for?

As long as you turn on NIC Teaming (at least that's term used in Windows Server for having all 4-5 NIC's bandwidth aggregated) and turning on in your switch Etherchannel on each specific port that will connect to your Server's NICs (Etherchannel being the term used on Cisco switches) then yes, you would have a 4-5gpbs bandwidth for your server.

If your switch is manageable search for any option similar to Etherchannel (something that will make the aggregate link between specific chosen ports).

Correct if he is only doing streaming outbound from the server and Internet file downloads are stored directly to the server as they download.

However, he can easily have peak transfer rate issues interrupting his streams. The real problem is movement of large complete files across the server connection -- NOT streaming of media which is inherently limited in transfer speed.
Although the link aggregation route can expand server bandwidth until the disk transfer rate is swamped -- several people doing uncontrolled swapping of large files through the server can still probably still make streaming media suffer jitter and stutter.

Especially as residential may well mean a frat house or other moderately large group. Peaks above 1GB of network link can easily happen if individuals dump large media files to the server from individual PCs when they get home (such files having been downloaded AFK via torrent during the day over several hours or days). Plus members may swap large media files via the server - often moving files locally either to escape interruption while watching on PC or loading it to personal mobile devices. People currently trying to watch media streams being served from server then get interrupted by LOCAL file transfers on the network. Even Internet streams could be interrupted if the switch backplane is overloaded at times.

Solution:

The quick & easy dirty man's solution is -- setup two servers: one for direct file storage and a second streaming only server connected by private "network" (crossover cable and two Ethernet ports) to access storage on the first server. Here uncontrolled file transfers can fight for access to the "public" network link while the second server streams over a clean switched connection. No managed switch required. As long as users have no direct SMB file access to the second media server then the only persons they can jam are themselves if uploading to the storage server while streaming from the media server. Even that may not happen unless they have a fast enough machine to process the media stream and go full bore on network file transfer plus any other UI or background tasks (AV software can slow them down quite a bit). Note: the two servers can probably be setup as two VMs on the same box though they each need to be assigned use of a separate real external network card with different IP addresses. The private network can be totally virtual or depending on the hosting VM service they may be able to share storage directly.

Alternatively you can set up two physical networks for the whole establishment with one local network dedicated to local file transfer and the other for local streams. If Internet download will leave adequate head room for everyone to run to stream locally you can put it on the local streaming network. NICs and cable are fairly cheap. Maybe all streaming is done on your wireless network if channels etc allow the bandwidth without congestion. The big issue is two connections to each computer mean an extra NIC and potential for Network routing loops (especially if any one hacks or plays with network settings they do not understand).

Or if you want learn something look at limiting or prioritizing use of the network connection to the existing server. Bandwidth quota, priorities, or quality of service (802.1p) at the server or switch (like Netgear GS108Tv2). Or if you can trust users not to change their machine its possible to set priority tags at client machines and have cheaper unmanaged switches respond based on those priorities.