I'm a network engineer at an ISP, so I would say I have a bit of experience with this from both ends of the table. First of all, there's a difference between your broadband connection speed and your perceived rate. Your broadband connection might be capped to what you pay for, and, assuming your last-mile medium can handle that speed, that only means that you will never actually go beyond your connection speed.
Now as we know, the internet is a complicated network of interconnected systems. You are connected via your ISP's backbone to the other systems (ISPs, enterprises, content providers, etc.) via a number of internet peering points. These peering points have their own connection speed (typically 1 Gbit/s or 10 Gbit/s, although higher exist), and may or may not be utilised to their maximum extent at any point of time. This means that you may have your full data rate available to some destinations, while others may take a congested route.
You mention testing, and your frustration is very reasonable. There are testing sites out there, but you never have any idea about how many else might be testing at the same time, or how much load there is on the server at the moment of the test. If you are unlucky, you might also be limited by your hardware, your operating system (TCP Window Size, receive buffers and similar might not be tuned properly), or your router.
I would say your best choice would be to download as much as possible from as many sources as possible (bittorrent is excellent for this, but may be throttled by evil ISPs), and do this over a couple of days to get an average indication of how much your connection is capable of delivering.
If you have a server on some remote location via the internet, you can use programs like iperf to make a bandwidth test, but such a test is not exactly precise when you have no idea how the intermediate networks are.