look through the literature (or find web-data) for your dsl modem/router, and see if there's a way to "talk" to it (mine is
http://192.168.1.1 -- which gets redirected; I'm guessing you don't have a (another) router hooked up to the modem/router: this can complicate things); if so, bookmark this address for future use. The modem (if you can "talk" to it) will typically report two data rates ("down"/"up"); and once you figure out what your normal rates are (these days, mine are 1182 (down) Kbps / 447 (up) Kbps), you can tell if the modem "thinks" there's a problem.
But sometimes there's some other problem with your ISP (or "upstream"), so run a speed test. I normally use
http://www.dslreports.com/speedtest?flash=1 . (The test-reported data-rates will usually be less than what the modem reports.) These days, sites are often so hooked together that a problem with one site (or a particular segment of the backbone) can have broad effects. (There are other speed-test sites (your ISP may offer one), and downloads from a reliable, fast site offer another alternative.)
I find that my data-rate from YouTube is highly variable, even when my link (to a test site, anyway) seems to be fine (this could still be a routing issue, not YouTube). And with my old computer equipment, I usually start a stream, then pause it and let it download completely before restarting the play. (There are issues related to computer and browser performance more generally that may be involved (eg, outdated sw versions); but this is a large topic -- and you'd expect to see these effects (perhaps in different form) more widely, at least under comparably high load.)
Turning the modem off, letting it rest for a while (my ISP says 15 seconds), and turning it back on often helps. (If I get really slow (effective) rates, often the modem is unhappy (commonly: reported nominal rates around 440; effective rates, much less), and this "reboot" works.)
And sometimes I get indicators that they're working in the CO again (where your phone line goes first, and where their dsl equipment is -- they tend to work there at particular hours; the modem gets disconnected and reconnected multiple times; I get weird data rates; the lights on the modem are different, change; etc). Plus, there's a known problem with acorns (and exposure to the elements... yes, acorns) in equipment down the street (or so I've been told); and so wet weather tends to create problems (they're supposed to be going to fix this).
When I have a problem that seems to be my ISP's fault, generally I just wait for it to be fixed. Their support is abysmal; they try to sell you a modem maintenance plan; and generally it's just a frustrating, pointless experience. But then, I think maybe that's the general idea.
It's helpful to have some idea how this stuff works, so you can at least figure out if you should try to do something.