With the newer-style openers these kinds of occurrences are becoming pretty rare. But it can still happen from time to time, since even the advanced coding systems still use similar transmitting frequencies for those codes and each opener is searching for a match on those frequencies.
You mentioned that both units had been reprogrammed but, unless the right precautions were taken, each could simply re-learn the wrong codes all over again.Have your neighbor erase the memory on his opener and then unplug it so there is no power his opener can use to search for signals at all. Now, while his is unplugged, reset the memory on yours and reprogram all your remotes and any keyless entry. Once finished, unplug your opener (so it now cannot search for more signals) and have your neighbor plug his opener in, reset the memory, and reprogram all his transmitters.Once all the programming is done on both units you and your neighbor should be able to power both openers up and use them with no further problems.If the problem does repeat then there may be something wrong with one of the remotes or openers and you may need to have a professional look into the matter further to determine exact what needs to be done or what options you have.The links below have additional information that should help for most newer-style openers and your Owner's Manual will have more programming and troubleshooting information for your specific model.