Well, while high stats help with most fish, rare "big fish" and quest-only types are generally just on a set RNG table.
Meaning, if you do have the minimum required stats for the fish to be in the loot-pool, and you have the correct bait/time of day/weather/location/active quest/any-other-variable that effects the direct ability to catch said fish, then that fish will be in the RNG loot pool but subject solely to it's fixed values and RNG.
The fish's "hidden stats" come into play of which we can only theory about it's effect, meaning, lets say for example, the target fish has a "hidden stat" of it has a 70% chance on a bite to get away, so no matter how high your FSH job level is over the fish's level or how high your Gather/Perception/GP is, 70% of the time it will always get away.
It gets really annoying on old ARR content "big fish" which have an on bite fixed value to get away (among other hidden stats), generally always getting away it seems, when you should be the god of the fishing pole having stats a couple thousand points over what is required to get a bite from one.

However, that is one of the reasons that the FSH job is closer to real life, as while stats help to unlock the fish in question, it really boils down to if the fish will let you catch it.