Layers separate because of moisture, excess adhesive, edge impact or poor shaping. Early service costs less than playing with a moving tip.
Fitting
Clean and flatten the faces, apply a thin compatible adhesive layer and allow full cure. Do not shape or hit balls before the adhesive has set.
Play and storage
Chalk evenly, never scrape the tip with a hard tool and store the cue in a dry case. In humid air, feel around the edge more often.
If a gap appears between layers, replace the tip instead of flooding the gap with glue.