Hi kittykatmax,
People certainly do double up mattress pads/protectors on their mattresses, but there can be issues with breathability and comfort impact.
If you go with a complete encasement that has a polyurethane waterproof membrane, while this is breathable, the polyurethane film doesn’t breathe as easily as a natural or polyester filled mattress pad or encasement. Placing a fitted-sheet style mattress protector over this will have a double polyurethane film on top, which certainly will offer extra protection, but it will also impact comfort and the ability of the top layers to conform. The breathability won’t be impacted as much by the addition of this second layer, as this piece will allow air to escape along the sides and bottom, only having a polyurethane film on the top. While the mattress would still breathe, the comfort would be more dramatically impacted.
If you placed two fitted-sheet style mattress protectors over each other, this would allow for more airflow around the mattress but would still provide a double layer of protection on the top panel of the mattress. This would impact comfort like the first option, but not quite as great, as the complete encasement would impact conformation a bit more that having the fitted as the base layer, as it would slightly firm up as air slowly escaped when weight was applied to the mattress. On the downside, you would lose the waterproof protection along the sides and bottom of the mattress (which might be important to your with cats).
Overall, doubling up two fitted polyurethane film mattress protectors would provide better breathability and impact overall comfort less, as compared to your alternative of a complete encasement with a fitted protector over that.
Phoenix