Is it possible to combine two internet connections to increase performance?

I've got a small home network, 3 PCs plus a laptop or two when the relatives come to visit, connected to a single cable internet connection. Now, as soon as everyone starts using the 'net the performance starts to suffer and if the load is heavy enough nobody can get anything done and everyone complains. At one point it was so bad that only one of us could use it at a time. I was researching possible solutions to this problem and I heard that internet cafes that utilize 2 internet connections, possibly from different providers, and have some sort of router that allows them to split the traffic between the both of them, with online games going through one and web traffic going through another. Is this possible? What is the technical term for it, and can/should it be applied to a home network setup or is there another solution to this problem?

There are several possible solutions.

1. Subscribe at your ISP to higher bandwidth - your bandwidth test results are very low.
2. If the bandwidth should be sufficient, then get a better router (remember to convert bandwidth in Mbits to Mbytes by dividing by 8, although dividing by 10 is normally more accurate).
3. If you still wish to combine two Internet connections, get a router that is capable of "Bandwidth Aggregation" or "Link aggregation".

For bandwidth requirements for youtube see :
How much bandwidth is required to play online YouTube videos ?

The article notes that to enjoy good HD viewing experience requires 1Mbps, while non-HD requires at least 513Kbps. Below these values, one may need to wait for the video to buffer .

I think your best bet is to look for a router with a QoS (Quality of Service) feature.

This will prioritise traffic going through your network and make more bandwidth available for things like games and streaming Youtube while reducing the bandwidth for things like Windows Updates, emails and other less time-critical things.

It's not going to work miracles, your connection is what it is, but it will make better use of what's there. This will save you a lot of time from manually going to each computer and setting Windows Updates to happen later or disabling background programs, and it will work for every computer on your network.

A still better solution though would be to install a separate internet connection for each computer, as another commenter said, although I don't imagine that is very practical.