My experience Andros, Biscayne, and Key West..agree 200...I prefer because I ain't as fast retrieving as in me 40 or 50 ies now just about 79... large diameter reels such as Nautilus Silver King 5' diameter for Permit and Bonefish so I will use as functional backing 200 if I need a filler under the functional backing to achieve a full spool then any old line will do. I fish primarily with a guide thus never more than 100-150 out...as for wading on Andros in my youth...if more than 150 was out the fish would find junk and cut me off or shark would find and eat...I always used at least 12-16 lb fluro and an 8wt...what I do now with my Silver King or Largest Danielsson HD 5 or Super 9/11 is use old braid as filler then change out with a new, every season, 150 yd braid spool of 65 lb on top to fill the spool when flyline is on top...my experience with all bonefish and all Permit and even most Tarpon after the first big run they come back to the boat and if you cannot keep tension with fast retrieve and then slack then probable hook fallout.
Another thing to think about- in many areas you may come upon a Permit. My first (and BY FAR largest) came while happily wading after 4lb Bonefish. I DID see my arbor on that fish- but ran out into chest deep water and eventually belly flopped into the guide's boat when he came motoring around the point with my buddy. Seeing 4 lonely turns of backing left while fighting the fish of a lifetime is NOT comforting.
The real answer to the question is not only about how much you need, but how much you need to fill the spool without joining the backing,which is why all my reels bar the smallest (Abel Super 6) have 300 yards of 50lb.(.012") Braid....and the 50lb.is more for handling purposes than anything else.A hundred or 150 yards won't go near filling a large arbor reel used for Bonefish or Permit.Additionally,as others point out,unless you're on a wide open sandy flat a Bonefish ain't a comin' back if he takes anything like a 100 yards plus the same length of your Flyline.
Reels include Tibor Everglades,Ross Evo R Salt 7/8,Abel 7/8N and Super 9/10.
As dynaflow correctly points out, backing is not merely there for long runs but for arbor expansion too. The reel's drag and your fish controlling technique are intended to preclude a bonefish getting excessive amounts of string off your reel.
Assorted Bonefish Reels post trip washed and drying: Evo R/S, Hatch, NV-G, Atoll and Omega