PA-28 Instructor reported receiving a GPS loss of integrity indication multiple times in similar areas resulted in questionable GPS signal or aircraft equipment.

Narrative

While conducting a VFR practice RNAV RWY XX approach into ZZZ on Day 2; my instrument student and I experienced an 'LOI' (Loss of Integrity) indication on the final approach into ZZZ. After receiving radar vectors to the final approach course and proceeding inbound; my student and I were near the final approach fix when we experienced the 'LOI' indication. Upon seeing this; my student and I executed a missed approach and began climbing. We contacted the tower alerting them that we were going missed and that we had an 'LOI' indication. Shortly after informing the tower; they told us to contact ZZZ approach control. Once we contacted ZZZ approach control; we regained our GPS signal and the GPS appeared to be accurate; so we proceeded to fly the published missed from where we went missed. After seeing the GPS signal return; we requested the RNAV RWY XY practice approach into ZZZ1; full stop. We did not experience any other 'LOI' indications for the remainder of the flight. Days earlier; on Day 0; I took this same instrument student to ZZZ and we did the VFR practice RNAV RWY XZ approach into ZZZ. A similar occurrence happened; this time; we were closer to the runway and on the final approach segment. On this approach; we experienced an 'LOI' indication from the GPS. Once receiving this message on the GPS; we conducted a missed approach and advised tower we were going missed and that we had an 'LOI' indication. Once we contacted ZZZ Approach; we explained how we had an 'LOI' indication on approach to runway XZ. The GPS signal was regained shortly after going missed; and it appeared to be accurate. As a result; we made another request for the practice RNAV RWY XZ approach into ZZZ1; full stop. For the remainder of the flight to ZZZ1; the GPS did not give us an 'LOI' indication. After thinking about these two flights; I realize that it seems as though the GPS abnormalities were unique to the area surrounding ZZZ. However; upon contacting approach; we should have informed them we would like to go back to ZZZ1 (where the flight originated) VFR; instead of requesting full-stop GPS practice approaches into ZZZ1. In the moment; I thought quickly that this was simply a ZZZ area issue and that the GPS would be 100% reliable for an approach back into ZZZ1. However; despite the GPS correcting itself and finding our location and course; I cannot determine for certain that the GPS signal was 100% reliable. Therefore; after receiving a GPS 'LOI;' I should not continue to use the GPS as a primary means of navigating or on an instrument approach. Moving forward; if this GPS 'LOI' indication is to come up again on a flight; I will continue to inform ATC of it happening; but we will then request to fly back VFR or by other navigational means to our destination airport; without another RNAV procedure.