The dock connector was completely corroded (covered with the green stuff you get when copper or brass corrodes). The poor iPhone kept thinking that something was connected, and charging was not possible anymore. This happened when I carried the iPhone on a snowboard trip in my pocket (sweat contains water and salt..)
It took a lot of time (1.5 hours disassemble, 2.5 hours assemble), but I stayed patient and the phone works perfectly now.
Take your time. Make sure you don't have appointments for the rest of the day.
I used a 10 inch piece of ducktape (sticky side up) to collect all the screws, with piece of paper stuck to it so I could write down from which steps all the screws came from. Very helpful.