Quoted: I've seen a lot of people spend a lot of money on insanely accurate rifles, and it brings a question to mind: Can you outshoot your rifle? Or rather, have you become so precise with shooting that the inaccuracy of your rifle is a real factor yet?

I have hundreds of hours of professional weapons training. I train regularly alongside phenomenal shooters and thus have an accurate understanding of my shooting skill.

I am well above average in the marksmanship department.

...and I have yet to encounter a situation where I was not the weakest link in the chain of weapon, ammo, and shooter, ESPECIALLY in dynamic shooting drills designed to replicate/approximate shooting under the stress and conditions of a gunfight.

That may not make my internet penis seem huge, but it's the God's-honest truth.

Now there are certainly some people out there who are not the weakest link in the chain of weapon, ammo, and shooter....but in all truth there ain't many of them.

The original question is poorly phrased (no offense).

A better question is of two parts:

1) What is the limit of precision of your rifle/ammo combination?

and

2) Can you shoot the rifle up to that limit without any external aids?

The first question is partially answered by shooting the rifle from a sandbagged benchrest on a dead calm day.  IOW, as many human and environmental variables as possible are eliminated or at least reduced.  Obviously the quality of the ammo plays a significant role, and this in itself is a subject of almost endless discussion.
FWIW, good quality general-issue milsurp ammo is about 2 MOA stuff, and there is a lot of ammo that does not reach even that level.  For simplicity's sake, Federal Match ammo with a bullet weight/configuration optimized for your particular barrel  is a decent benchmark, but a skilled handloader can do a bit better, maybe considerably better after a bit of experimentation with a particular rifle.

The second question can be answered only after the first question is resolved.  At that point, it is your level of skill that determines how close you can approach the rifle's established capabilities.  That means shooting from various positions, and not from a bench or a bipod.
While you should know if there is a POI change when using a bipod or benchrest versus shooting prone, for example, your prone/sitting/kneeling/offhand position shooting will reveal your actual strengths and weaknesses.  Sandbags/benches and bipods remove at least some of the human variable, and thus can be deceptive in assessing a shooter's skills.

Having said the above, it is my experience that very few shooters bother to expend the time, effort, and money required to hone their skills to a point where they are anywhere near shooting to their rifle/ammo limit of precision.

It is sad to say, but true nonetheless.