While shutter speeds are fundamental in photography, it is possible to get over-excited about them. With negative film, the ill consequences of overexposure are minimal: slightly bigger grain with conventional B+W film (slightly finer with colour or Ilford XP-2) and slightly reduced sharpness. Most people prefer the tonality with a little extra exposure anyway. With slide film, you can simply use a higher marked shutter speed: one of my Pentax SVs runs at about half speed from 1/60 to 1/1000th.
Also, tolerances on shutter speeds are much bigger than most people think. Traditionally, speeds of 1/1000 and above are allowed to be +/- 2/3 stop, with other speeds at +/- 1/3 stop. That's 1/1300-1/2600 at 1/2000; 1/640-1/1600 at 1/1000; 1/400-1/640 at 1/599; 1/200-1/320 at 1/250; 1/100-1/160 at 1/125; and so on down. They rarely run fast.
Most cameras are better than this when they're new, but after a decade or two or even five or six, they tend to slow down. As I have an old ZTS Tester PRO (no longer made), I can actually test them, and recently I did so when a young friend was trying to sell five Leicas. The results were illuminating, and not a little depressing: you can see the actual result in the linked article.
Shutter speeds can vary according to the orientation of the camera, i.e. whether it's held "portrait" or "landscape" or indeed looking straight up or straight down. This is nothing unusual: mechanical watches are traditionally tested "crown up", "crown down", etc. Also, you sometimes get different speeds according to which direction you are coming from, i.e. if you go from a slower speed to a faster one, you may find that the speeds are not identical to the results you get when going from a faster speed to a slower one.
If you don't have a shutter speed tester, and don't feel like buying one or building one there are at least two other ways to test the shutter more or less accurately. One involves photographing a television screen and counting the lines, and the other involves photographing a rotating record turntable. Both are described here (RIT) but be warned that they refer to the American NTSC standard: you'd need to look out the other standards for other countries, and frankly, it all looks like more trouble than it's worth. Given what I've already said about a pragmatic approach, there are two much simpler tricks that are worth knowing.
First, listen to the shutter -- but listen to alternate speeds. Adjacent speeds often sound very alike, but the difference is usually much clearer if you listen first to 1 - 1/4 - 1/15 - 1/60 - 1/250 - 1/1000 and then to 1/2 - 1/8 - 1/30 - 1/125 - 1/500 and (if you have it) 1/2000.
Second, with the back of the camera open and the lens off (or at full aperture if the lens is fixed), just fire the shutter at all speeds. Sometimes you can see the shutter hanging up, either with the leaves opening or closing slowly or partially, or (in the case of a focal plane shutter) one shutter or the other sticking at some point in its travel, whether open or closed. You may also also see, at the very highest shutter speeds with focal plane shutters, that the shutter isn't opening at all: the second blind is released at the same time as the first, and they travel across together.
If the shutter is working at all, remember that like most of us as we get older, shutters tend to be happiest at a comfortable room temperature, and they may be even happier in a warm or even hot room. If they're too cold, on the other hand, they may slow down or stop.
And, again like most of us, they respond well to exercise. Repeatedly winding and firing a shutter will often make it work better, though admittedly at the cost of very slight increased wear as we operate it. Given that most shutters can handle hundreds of thousands of exposures, it is possible to overestimate the importance of this.
Leaf shutters in particular are very easy to bring back to life this way. It is usually the slow speeds that stick, so start with the higher speeds and work your way down. Usually, the slow speed train engages at anything from 1/40 to 1/15 second, depending on the shutter design. If (say) 1/10 is working and 1/5 isn't, repeated exercise of the 1/10 may bring the 1/5 back to life. If it does, repeated exercise of the 1/5 may bring the 1/2 back to life, and so forth. Also, VERY gentle assistance of the cocking lever (with a finger tip) may persuade reluctant slow speeds to reach the end of their travel. At this point, with elderly, cheap cameras, it becomes a question of "what have you to lose?"