When Thomas Anderson (Neo) is waiting for Morpheous' team to pick him up under the Cherry Street bridge, it is raining really hard. However, look at Neo, he is completely dry, even though he has no umbrella! How was he able to walk from his apartment to a bridge in a downpour without any protection, and remain dry?!
I don't know about you, but he looks pretty damp to me.
My explanation is that he walked there, in the rain, and then had to wait for a long time for the car to show up. He's under the bridge, so he won't get wetter, only drier. So when the car shows up, he's almost completely dry!
1. It might not have been raining when he walked.
2. He might not have walked at all.
3. Standing under the bridge long enough to dry.
4. He is damp and not wet.
This was done as a forshadowing effect. Neo is the one, however at this point in the movie, he does not know it. At this point, he staying dry, was done because he is great. And he knows kung fu!