It does not harm a fan for you to stop it for a few seconds. By that I mean use a pencil or something similar and use the eraser end to run against the spinning fan blades gently to stop it for a few seconds, then release it. This is handy for testing things.
What that caution is about is that if you send the fan a signal to run too slow it will stall, which has two results. The obvious: no cooling from a stalled fan. Less obvious: the fan still has a small current running though its windings and generating a little heat, but it cannot cool itself. The way to re-start it is to supply a signal to run quite a bit faster. On older 3-pin fans, the "signal" is just the VOLTAGE supplied from Pin #2 of the header - reducing that voltage is how that fan type's speed can be slowed. Most fans of that type will stall if Voltage falls below 5 V. On new 4-pin PWM fans the Voltage supply is constant, but the PWM signal from Pin #4 tells the fan what speed to run. That, too, can be set too low and cause a fan to stall, although that may happen at a lower minimum speed than with an older3-pin fan. Again, the fix is to send it a higher-speed PWM signal.
On virtually all mobo fan headers when you allow it to use an automatic fan speed control mode it also monitors the actual speed of the fan. IF there is NO speed signal (or in a few mobos, if the speed falls below a minimum you can specify) the system calls that a fan failure and the first response is to send out a higher-speed signal to re-start it. If that works, the automatic system goes back to what it was doing before. BUT that may have been the cause of the stall, so it may happen over and over again. If the fan does NOT re-start you get a warning on your screen of the fan's failure.
Most fan headers also allow you to set a fixed speed and NOT use the automatic systems. In that mode of operation the header may NOT monitor for fan failure and do an automatic re-start. That is one way you can cause a fan to stall and not have it corrected for you.
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