Like many others said, without a tripod there’s usually (for exceptions see below) some blur. But if you can’t use a tripod, than this rule can help a lot. But, it is a rule for 35mm SLRs only (not sure if it can extend to MF SLRs too). The fact is that the slap of the mirror is the main cause you cannot go for longer exposures than the rule suggests. In the second place there’s the weight of the lens – with a heavier telelens if the camera can hold it in console (so you only touch the camera with your hands), you could go for double the exposure of this the rule. From my experience, it worked up to 300mm.
But, if the telelens is too long, the rule doesn’t apply anymore. I have a 600mm reflex (old Sigma which originates from Ricoh) with a tripod mount on it, but I couldn’t use a tripod mounted to the lens, even with mirror lock-up, because the shutter’s blades made it vibrate even at 1/2000s. So, my only chance was to use it in console, but it was too heavy for the camera to hold it this way. Finally, I took of all the tripod mount system of the lens, and it become a lot lighter, so I can use it now in console. With the camera handheld, I can go now up to 1/250s with very satisfactory results, and with some luck I can go even to 1/125s. But if I mount the camera on the tripod, with the lens in console this time, even using the mirror lock-up, it still vibrates from the shutter blades. And I have a tripod made for MF. It seams that handholding is far better than using a tripod in this case. So, none of the rules mentioned in the beginning does apply for this combo, but their opposites work very well.
Now, if I use a 50mm lens, I can handhold a SLR up to 1/30s. If I use a 28mm lens, I can handhold the camera also up to 1/30s. It seams that below the normal focal length the rule stops to apply either. Here, the exposure time is given by the capacity of the camera to absorb the mirror’s vibrations by its weight, the bumper, the mirror’s weight and acceleration. Although, my hands can stay still mostly up to 1/15s with normal lenses – see below.
Another example: the range-finder or the TLR with leaf shutters. With these, I can go down to 1/15s with very satisfactory results. I cannot go below because no matter the lens length, or the camera weight, my hands cannot stay still for longer laps of time. No rule applies here, other than my hands rule.
So, these rules, even that one recommending a tripod, are only approximations. But again, the rule considering the relation between the focal length and the exposure emerged for 35mm SLRs equipped with lenses starting at 45-60mm and (I would say from personal experience) up to 300mm (and less for wideangles).