How to test internet connection quality over time?

I find pathping to be a very useful utility in these cases, at least if you're on windows. I'm currently on linux, so I unfortunately do not have the ability to verify the syntax, but if my memory serves me right, it's either similar or identical to that of ping.

Basically, it works like ping, but instead of pinging just your target, it also pings every hop of the way. It does this by first doing a traceroute to figure out each hop, and then it'll start pinging each of those, and keep doing so for 10 minutes to gather statistics. This way it'll be able to see if there's an obvious problem somewhere. Source: An ISP once asked for info from pathping when troubleshooting my SDSL once upon a time.

It is worth noticing that ICMP-ping is very limited, so it won't put any stress on the bandwidth, and in most cases, problems like these only show up when there's a significant traffic. Try shoving a lot of date to or from an FTP server or something similar, if possible, one that you can also pathping.

As for utilizing your server, this could possibly be the FTP mentioned above (it doesn't have to be FTP... any protocol will do, really). Additionally, using nc if you're on a unix-oriented system is also a good way of finding additional information for bandwidth behavior.

You could use smokeping: http://oss.oetiker.ch/smokeping/
It will draw graphs, were any fluctuation and drops in your connectivity will be visible.