Umm, be careful. My first thing to do would be to get one of those lead solder test sticks, which will tell you what type of solder was originally used.
If leaded, then fine, go ahead.
If unleaded then the fun starts. Using leaded solder on an unleaded joint will, as scientist and eKretz say, be a mess. What seems to be the problem is that the joint ends up as a 90/10 leaded solder joint, and that amount of lead doesn't work. This is why cleaning all the old solder off is important, so that the joint is actually 60/40. Apparently you can get the same problem by simply using a leaded solder iron with lead free solder, disaster.
An alternative might be to use a small hot air pencil to reflow the joints, use a dab of flux on them as well.