I can imagine a situation when you are re-cleared from ILS to a non-prec approach that uses a different ground track, e.g. a VOR approach with VOR off the centerline. (sorry, can't remember how are those properly called).
Your airplane would not fly the VOR app track, but would try to intercept the ILS loc, even if you only planned to use that for reference. If the VOR approach is angled some 5° off the LOC, you can already run into quite a bit of distance error mid approach.
In general, I don't know why you would fly a VOR approach if the ILS was operational. If for practice, Parking the ILS would be a good option.
The example I'm familiar with is Rwy 22L at JFK. The RNAV approach overlies the ILS localizer with a 225 course, but the VOR comes in at 232, crossing the extended centerline near the MAP. If the ILS (or G/S) was inop, the RNAV approach would be a better alternative than the VOR. However, I have gone into JFK when only the VOR approach was available, for whatever reason. It's been a while, but I suspect the ILS was out of service, so capturing the LOC would not be an issue.