I think the advice contained in this thread is good. However I just thought of something nobody has mentioned. Occasionally in repairs, you will come across a component that was made using aluminum or steel wire. Both of these require special processes and are difficult or almost impossible to re-solder using normal rosin core solder. There are special fluxes for aluminum ( at high temperature, I think ), and steel I believe is soldered with silver. These materials can look like they have been soldered normally, but actually a special process was used at the factory.
My guess is that you are NOT running into this, but it is a possible explanation when you run across a wire that simply will not solder normally.