Patches, entities (including mapmodels, func_statics, doors, and such), and brushes explicitly set to “detail” in Radiant (ctrl+M, or menu option). The latter have no particular limitations, other than that they will not cull pvs or block leaks. So if you made your map entirely with “detail”, you’d leak; if you threw a caulk box around it (not made detail!) it would quit leaking but the whole map would be drawn from every vantage. Normally, one puts non-detail (structural) caulk cores in the floors, walls, and ceilings and behind skies, etc. to keep visibility under control. The map then has this simple boxy layout of caulk, with a crusting of textured detail faces.
Overdraw prevention is the other major area that gets affected – bits of texture behind or on the back of detail brushes may still render, unlike on structural ones. It’s best to caulk every square centimeter of unseen geometry regardless.
There are minor problems that can happen as well: detail brushes can cause areaportals to leak (put structural caulk in all the walls, floors, and ceilings); detail brushes can cause light to leak (if you get light leaking under a stubby wall, for example, make it structural or put structural caulk inside it and look into hinting the areas on either side); and detail brushes may cause fog volumes to act up – to be safe, fog should be enclosed on five sides with structural brushwork.
This construction, done right, will speed up vis times drastically and avoid all the vis stage errors, notably MAX_MAP_VISIBILITY and the one you’ve hit, without significantly worsening framerates or polygon counts.
Note that brushes that use a shader with surfaceparm trans (notably, clips and such, and see-through grates, flames, and similar sfx) don’t block vis and may count as detail. (I’m unclear on whether surfaceparm trans suppresses the brush from producing bsp splits or portals, or just from being considered opaque by vis. They definitely can cause leaks, with the whole map or with areaportals.)