A friend told me that when the time signature is 4/4, if your playing is not in sync with the other band members, then you can solve the problem by a passage. But this cannot happen if the time signature is 6/8.
Is this correct? Why?
I think I know what you're getting at. First, a "musical passage" is just any group of notes that are organized into a single idea. It doesn't have anything to do with being on the beat or not. However...
If you've lost the beat, you can stop playing, get back onto the beat, and start back in again while everyone snickers at you. Or, you can play a number of rapid notes to cover the fact that you've lost the beat (one or another of those notes is going to fall on the beat), and under that cover, find the beat again.
You can do this in 6/8 as well, but it's harder. 6/8 is usually a fast tempo, so if you want to cover losing the beat, you have to play a run (or passage) of even faster notes to cover it. If you can play 16th notes at the tempo, you should be able to pull it off, though.
Instead of trying to cover up timing errors with extra notes, you may be better off stopping, listening, and re-starting. In those extra notes, you're just as likely to get out of time again. 4/4, 3/4, 6/8 will all come down, in the end, to being capable of providing this 'passage', but it's like a pill to cover up the symptoms - it doesn't solve the problem - you find it hard to keep time. If 4/4 is tricky, then others will be worse.
Get used to playing, by yourself, with a metronome, or backing tracks. Try different tempos, feels, time sigs. Of course, it may not be you going out of time. I've played with drummers who are anything but good time keepers - technically brilliant, maybe, but not actually doing their basic job - time keeping (in most numbers, be fair !). Record what is being played, and listen back critically - you will probably find where things start to go out of time, and it may not even be you that instigates it.
Basic answer to the original question - it's a poor man's idea of getting out of the mire, and any time sig. will not help.