I am referring to the stage where someone reaches his/her capabilities limits, especially professional limits. I am thinking about career advancements as well as the growth in professional life. We all reach our limit in that progress and going further would be unnatural or might put us in a difficult position because we are not able to cope with a higher level of abilities or responsibilities. Is there a word or expression to define this limit in personal progress?
Medica's answer pointing to the Peter Principle is on the money, however just to add a phrase or two...
colloquially: maxed out, plateaued, sweet spot
Having "risen to his level of incompetence" refers to having gone beyond Jack's referred to territory of competence. "Out of his depths" would similarly express having risen above that territory.
Actually, you are describing the Peter Principle to a "T".
The Peter Principle is the principle that "in a hierarchy every employee tends to rise to their level of incompetence".
Formulated by Dr. Laurence J. Peter and Raymond Hull in the late 60's,
The Peter Principle is a special case of a ubiquitous observation: anything that works will be used in progressively more challenging applications until it fails. There is much temptation to use what has worked before, even when it may exceed its effective scope. Dr. Peter observed this about humans.
[Similarly] members of a hierarchical organization eventually are promoted to their highest level of competence, after which further promotion raises them to incompetence. That level is the employee's "level of incompetence" where the employee has no chance of further promotion, thus reaching their career's ceiling in an organization.
The Principle further states that the work of that person who has reached their level of incompetence tends to be done by those below him who are better equipped. Managing upward is the idea that a better-suited subordinate finds ways to subtly "manage" superiors in order to limit the damage that they end up doing.
The employee's incompetence is not necessarily exposed as a result of the higher-ranking position being more difficult — simply, that job is different from the job in which the employee previously excelled, and thus requires different work skills, which the employee may not possess. For example, a factory worker's excellence in their job can earn them promotion to manager, at which point the skills that earned them their promotion no longer apply to their job.
On an very public level, the disastrous response by FEMA to Hurricane Katrina was shown to be the result of FEMA Director Michael D. Brown's being a victim of a poor promotion. Brown had excelled in his previous position, and naturally was given a job with greater responsibilities by President G.W. Bush.
Brown was vilified by the media, but it's difficult not to commiserate with him. He was good at his previous job, and -- as is dictated by the American Dream -- when offered a position with more prestige, salary and potential for growth, he took it. The person who gave him the job had faith in his abilities, so why shouldn't he take the job? But Brown proves that a promotion isn't always a good thing.
This seems to match well your statement that "We all reach our limit in that progress and going further would be unnatural or might put us in a difficult position because we are not able to cope with a higher level of abilities or responsibilities.
Whether it has a more succinct or different name, I don't know. I've only ever heard this referred to as the Peter Principle.
You might describe it non-specifically: you've exhausted your potential, reached your limits of expertise, you'll be in over your head, etc.
Fulfilled your potential
That might be the most neutral, least provocative way of expressing this idea.
maxed out and plateaued
Are good suggestions from @MarkKasson, although they may be slightly pejorative.
Also from @MarkKasson's answer, is not correct IMO. This expression has nothing to do with reaching a limit - it simply means you've gotten something "exactly right"; "the stars were perfectly aligned"; "everything came together".
I am thinking about career advancements as well as the growth in professional life.
The Peter Principle is way off. You are describing a situation where you want to progress, right? Also, you say that it is 'personal progress'...?
If you are concerned about your own personal abilities to become a person who can learn to advance, or adapt to the demands of being more creative/organized/productive, then that is a personal development issue. You might be frazzled at your limit, and raising the bar might make you feel worse, as you would feel you couldn't perform at 100% even with your best efforts...
If you are concerned about your career being stonewalled, or hitting a dead end, or any of the other terms, then it is a matter of being the kind of person that will fill any void of opportunity. That is, you will get bored/sloppy/lazy if you go to work on auto-pilot.
I see a lot of folks stay in one position forever. They have a job. Others are completely unsatisfied where they are, and are continually moving up... They have a career.
Avoiding the risk of pissing off the moderators for ending an answer with a question mark (however professional/polite it is):
Ask yourself what type of person you are, and you will know what kind of professional you are.