Trombone players have become more gear obsessed as they've had more custom options available in recent years (the double-valve tenor is the most egregious example--you might as well play valve bone), and they can be really gearheaded about mouthpieces, since trombones are relatively simple instruments and there isn't that much to customize, comparatively, but I think Dave Wooten will back me up if I said I thought trumpet players were bigger gearheads than trombone players.
A normal 'bone player can get by with one horn, but it's pretty typical for a proficient player to have two, three, or four horns (for the record, I have three)--maybe a small bore for jazz, a larger bore for orchestras and other classical ensembles, maybe an alto or bass or a valve instrument in the same key like a euphonium, baritone horn, bass trumpet or valve trombone or a historic instrument like a sackbut, contrabass, or serpent. Occasionally you meet someone who likes to collect horns and has more. Trombones aren't particularly valuable compared to other instruments--professional quality instruments being in the range of $1000-4000 new and used, stock and custom (maybe more for something like a contrabass or a reproduction of a historic sackbut, but these are fairly unusual). Then people try several different mouthpieces until they find the right one or two or three, and maybe they sell the old ones or trade them in.
Trumpet players--I don't know. I guess it starts in the orchestra where they might decide they need a piccolo trumpet, C, D, and F trumpets in addition to the regular B-flat trumpet, and then sometimes it's stylistically appropriate to have a cornet or a rotary valve trumpet, and then you gotta have a classic New York Bach, and there's something special about Schilkes, and maybe something moderne with sheet bracing, and wouldn't it be fun to have a pocket trumpet, and before you know it you've just got dozens of them, and then it's mouthpieces--heavy weight, skeletonized, cushion rims, thin rims, shallow cup, deep cup, drilled throat--and then it's little tweaks--heavy valve caps, extra bracing, lacquer stripping, gold plating, it just never ends.