Vibrato units rely on springs which counter-balance the strings. When set up normally, everything's in tune. However, when a string breaks, there is less tension on the string side of the balance, putting the other five slightly out of tune, as the spring now has more tension compared with the strings.
The opposite will happen when a string is bent. This causes more tension on the string side, so the other strings will have slightly less pull against the springs. Thus they'll drop in pitch, albeit very slightly.
It's one of the downsides of vibrato systems, and can be countered by bending the offending string up to pitch - which then makes all the others even worse!
It could be the neck, although on quality guitars it's rather unusual.