"Does anyone know what is wrong when a guitar, tuned using an electronic tuner is then 'off' when the strings are checked against eachother?"

First, it often depends on which chords you are playing. Electric tuners are almost pitch perfect. Chords are 'relative' in the sense that a guitar can be in tune for the C chord and sound slightly off when you change to play in the key of E (for example). It is possible that your frets are not true or that there is a slight bow in the neck. Also, as you said, the bridge (saddle) may have worn or moved a bit. What make of guitar is it?