Well actually to be correct you should say:
"The runoff from heavy rains washed away the bridge." It is unlikely that the rainfall itself would wash away the bridge but the flood (runoff) that follows the rain could.
To say that the runoff from heavy rains washed away the bridge does indeed imply that it was the result of more than one rain event (IMHO) however if the rain is indeed continous it would be perfectly acceptable to say: "The runoff from a heavy rain washed away the bridge."
