I just had the same problem, but I wanted to let everyone know what the solution, and what the possible problems WERE.
My phone's only 6-8 weeks old, and I didn't get the insurance, so this was stressing me out. LOL
Last night, during the rainstorm, my phone slipped out of my jacket pocket, into a puddle and was submerged for like, 2 seconds.
I immediately wiped it completely off, hoping it was going to survive (which it did, thank god)... but my battery was low (I charge my phone overnight every day), and when I put it to charge, it wasn't realizing that it was plugged in. 0.0
When I wiggled the cord around, trying to make it pick it up, it eventually did say that the charger wasn't compatible to my phone......... which pissed me off.
It wouldn't charge AT ALL. ALL night.
I tried again this morning, and instead of using it in the wall, I plugged it into my computer, and after wiggling it again a lot, it finally picked it up, and it charged (very slowly, of course, since it was charging through the computer) a few percents, but then it stopped. And then the only activity I could get out of it was the "this charger isn't compatible with this phone" so I brought it to work with me just now, looked up on Google "Galaxy S4 charging problems" and found this forum.
Quickly skimming through answers, I found the idea to gently adjust that tiny silver piece inside the phone, in case that makes the charger take it.
IT DID WORK. I uncoiled a small paperclip and used the tip to wedge (GENTLY) the silver piece up a little ((so it's more centered than toward one side---- but not completely centered)), stuck my charger in again, and it ******* works! It's charging right now, and this literally happened less than 5 minutes ago.
So good luck to you guys!
((If you have a heavy hand, or are unable to be extremely gentle, then have a girl do it for you, if she's that way.--- You really don't want to **** this up. It's like playing Operation.))