Ok that explains everything!
Fair warning, large block of test. Feel free to skip it till the
bold line.
We're starting with theory.
The transistor is acting as a voltage regulator and the knob sets how much current is being fed to the base pin, which in turn sets how much current is allowed to flow through the transistor as a whole.
This makes calculating how many fans can be used an interesting and somewhat difficult exercise.
In theory the transistor is rated at 3 amps, assuming I'm looking at the correct unit (
datasheet). However it is also rated at 12.5W assuming a junction temp of 25c.
The transistor is acting to drop the voltage, at full on I expect a voltage drop of around 0.7v to 2v.
If you have fans that magically draw one amp at all voltage levels (lower voltage they'll draw fewer amps, that's part of what makes this awkward) and the transistor is dropping 1V at the full speed setting you calculate the power disapated by the transistor thusly:
Voltage drop (1) x amps (1) = watts (1).
A typical temp gain for that package is 100c per watt. If your case is running at 35c or so (the hot end of cases) and the transistor is getting little to no airflow that means that in the above example it'll be running at a toasty 135c. It's rated for 150c though, so that's OK. In reality the hotter it gets the better it conducts and the lower the voltage drop.
Now if you turn the fans down a bit, say from 11v (max) to 9v, now you have a 3v drop. If you still had 1 amp of current (like I said, you won't) you'd have three watts and the poor transistor is looking at temps of 335c. Poof! Dead.
In reality that theoretical 1a @ 11v fan would be drawing more like 0.8a at 9v, but that still puts the transistor way too hot.
More airflow helps keep it cool, as does a heatsink.
The part itself is rated for a maximum of 12.5w in any situation.
The above is all the theory, give that theory I would say 0.33a of fans at most, 0.25 amps max would be safer.
If you added a heatsink and airflow you could use probably twice as much fan.
How about some Real World? This is what I recommend you actually do.
Plug a fan in, any fan you know the draw of (it's on the sticker). Run it at full blast for a while.
Measure the temp of the transistor.
Turn the fan down, run it a couple minutes, check the temp again.
Keep doing that.
If the temps are reasonable, try a beefier fan or two of the first fan and re-test.
Keep in mind that 150c (max safe for the transistor) is plenty to burn flesh instantly. A fingertip isn't the worlds greatest thermometer for this.
Worst case situation the transistor dies. You can get a new one for $0.73 at
digikey.
If it dies it may smell pretty bad and make some deliciously nasty smoke while it does it.