Bathroom exhaust leaking when it rains
How long ago was it replaced or touched? E.g., roof repair/replaced, or energy efficiency program teams, or a contractor/self replacing the fan or the piping or exhaust hood?  I ask because the length of time may warrant further inspection to address any long term water damage issues. It could require new plywood (especially if you are in a cold snowy climate). I'm not saying this is the case, just making sure you give it some thought and inspect the damage areas closely.

If there is no long term damage a few things can be amiss....


Incorrect installation 
Flashing damaged/dried out
Vent fan hood damaged/worn out
uneven pressure in the home via some other exhaust/pressure source


In reverse order for 1-4:
If the home has positive pressure (e.g., a other vents vent, a window fan or other fan blowing IN, or even sealed so tight doors opening and closing), these things can blow the door open, allowing rainwater to blow in the vent.... gravity wins.

If the vent apparatus on the roof is damaged, ie... missing flap, bad spring, debris, bad seal, etc, it will allow water past the opening, and, again, gravity wins.

As Jasen noted, flashing that has dried out, or wasnt applied properly, or wind/debris damage bending the topper or surrounding tiles, even higher up the roof near the ridge, can allow water to travel (via gravity) to your vent opening. This can even happen with expose nail holes, screw holes, abandoned satellite dish holes, etc.

Lastly, installation - this can and is often the culprit, and the damage probaly has been happening all along (possibly). Make sure the vent tubing is the right size (not longer than needed) make sure it slopes properly to allow vapors out without giving a place for water to pool or get trapped. Make sure is it properly sealed/taped at each end, and insulated if in a colder climate. Use rigid pipe instead of plastic or aluminum ribbed ductwork (like most kits have). Also, make sure the direction of the topper makes sense, and ideally is not directly in the souternly path of wind/water.

Follow that list and you should nail it.