This question came up in a world and I can't remember the answer. Here's the situation: I have a low scouting budget but see a pitcher's stamina projections to be equal to his current stamina. Even though my scouting is low, am I right to assume his stamina will never increase, because a player can never exceed his projections?

I thought I remember being told that even though some spend more on scouting budgets, this doesn't mean that every category is more accurate than someone with lower budgets and that it's more of a general average.

No, your assumption is not correct.

What you see does not impact what a player can aspire to. He has limits on his growth (assuming no DITR) but those limits do not necessarily have any correlation to what you see.

What you see (your scouting) is less likely to be accurate (read: the same as his actual limit) than what someone who spends $20 on Adv Scouting sees, but neither of you is assured of accuracy with respect to any particular player or any particular rating.

+1 to what zbrent said.

FYI . .  when a player's current rating exceeds what you see as his projected rating, his projected rating will increase to match the current.  So in general, you'll never see a current rating exceed a projected rating, with one exception.  The exception being when a pitcher drops a poor pitch.  The poor pitch still shows as a projection, but not as a current.  In that event, you'll see a higher current overall rating than projected overall rating.

Also depends on how many years he's been playing. If he's got 4 years of experience then his Stamina likely will not increase any more, or if it does it will be only a point or two.