Assuming light leaks from the sides are taken care off with proper sealing, could there be light bouncing back and forth between the filters?
Yes. Just as lenses with more elements are potentially more subject to flare and ghosting than lenses with fewer elements, the more surfaces you add to an optical system, the greater the chance of increasing reflections between the various surfaces. With filters this is usually exacerbated by the fact they are flat and parallel to one another.
How much light is reflected by each surface in an optical system will be determined by the physical properties of the material used to make the element with that surface, and the physical properties of the anti-reflective coating(s), if any, that may be applied to the surface(s) of that element.
In the film era, coating the front of several optical elements in a lens was usually enough. With the higher reflectance of near perfectly flat digital sensors and the near perfectly flat filters in the stack immediately in front of them, it has become commonplace with higher quality lenses to coat both sides of almost all optical elements that include an air/refractive glass interface.
Most quality filters are multi-coated on both sides. But keep in mind that having a 'multi-coated filter' doesn't always mean that both sides are coated. 'Multi-coated' references the number of coats on one side of a filter. Both sides of the filter are not required to be coated to be considered 'multicoated'. Two or more coatings on one side of the filter are enough to make a filter 'multi-coated'.
If you have square/rectangular filters with anti-reflective coatings on only one side, you can experiment with turning them around to see which way produces fewer/weaker reflections. Of course there are some filters, such as circular polarizers, that only work in one direction. But most ND and color correction filters can be used turned either way.
As always, flare and reflections will be most noticeable when there are strong light sources in the frame or just outside the frame but allowed to fall on the filters or elements at the front of the lens. Shielding the front of the lens from strong off-axis light sources will go a long way towards reducing flare in most situations if there aren't also strong highlights within the field of view.