Brazilian Jiu Jitsu has a very clear system of progression. There are 5 adult belts and the IBJJF has guidelines for the minimum time required at each belt before an instructor can promote. Promotion is done at the will of the instructor and usually belts are awarded based on a combination of skill level and time spent on the mat.
Belt promotions can be a very emotional experience for some as it is confirmation and reward for the time and effort spent on the mats. If a belt is not awarded many people are left disappointed and left with feelings of frustration as they think that they have not been given the reward that they feel they have earned. People can easily become dejected as they can see their teammates progress while they remain static at a particular grade.
More rare is the student who does not want to be promoted. In this case the student may have spent the requisite time on the mat and be competent in the eyes of the professor but they still feel that they are not ready to progress to the next grade. There have been countless stories of students turning down promotions and handing back belts after a grading ceremony. I have witnessed this in my own academy and in my opinion there are two main reasons why someone may not want to be promoted.
This is the most common reason for not wanting to be promoted. These students feel like they do not deserve the belt and are reluctant to receive a reward which they feel is unearned. These students may not have strong self esteem in general and will often see themselves as inferior to to their team mates.
In general Jiu-Jitsu can be a great confidence builder as it is a excellent way of meeting and interacting with people while also providing physical fitness and providing a type of mindspace where people can forget about the stresses of everyday life. Despite this some students will still struggle with confidence and this is where a good coach is often required to step in to provide some reassurance.
Fear of losing
The fear of being exposed or consistently beaten by better grapplers is another common reason why some student try to avoid promotion. Losing consistently and being dominated by belts of the same color or lower than you can be hard to take and it is often enough of a reason to quit for many people. By holding onto a belt and not progressing to the next level, the student is protecting from this feeling. The truth hear is that each BJJ player is own their own journey and that being beaten by lower grades is not something that should leave you feeling ashamed. What is important is that you keep training and continue to working on fixing holes and developing your game.