I once heard that the grandmaster of a school of martial arts has the highest rank in the whole world. No one else can earn that rank, so when the grandmaster passes, the man or woman who has the next-highest rank becomes the grandmaster. However, even then, that person never attains the same rank as the previous grandmaster. The highest rank is simply lowered by one. Perhaps, I am confused, but would that mean that after many years, the highest rank would be (for example) a blue belt?
Martial Arts Traditions Vary
First, there's no universal structure for all martial arts. Different styles, different schools, different teachers within the same style, might have different ranks, or no ranking system at all. (Wikipedia on martial arts grandmasters)
Many styles (but not all, maybe not most, when you look at things globally), have a head of a style.
In some, when the head instructor passes, it goes down to the next highest ranked person. In some, a group of the most advanced instructors decide who is next. In some, the head instructor has nominated someone as an heir - usually either their most favored student or a family member.
In most cases, that person takes on that same rank of grandmaster/head of the lineage. Sometimes, they may get the top role, but if it has a belt ranking, they may still have to put in time or earn a higher rank as well.
Of course, there's also many situations where someone breaks off from a lineage, creating their own new lineage - changing the name, perhaps changing or mixing in other styles, or perhaps not at all - and then take on the role of "grandmaster" for their new lineage. This happens a lot when you have friction and political problems within a lineage, and in places where it's profitable to trademark a style and set yourself up as the person with the rights to teach these materials and license use of your school's name and logos.
Maybe the person who told you "never attains the same rank" might have somehow deeply confused a basic idea of Confucianism?
In Confucian traditions, there's a deep reverence for family, elders and teachers. So, if you were to inherit the title of Grandmaster or head of a style, you'd be the official top ranked person, but you'd still always pay respects to the instructor(s) who came before you. They would, at least in your esteem, "outrank" you.
However, Confucian philosophy and practices only applies to a subset of martial arts, and as you can see, even then doesn't affect or adjust the actual ranking of the living practitioners.
In judo 10th dan is the highest rank anyone can receive. Wikipedia attributes the following quotation to the founder of Judo, Jigaro Kano:
There is no limit...on the grade one can receive. Therefore if one does reach a stage above 10th dan... there is no reason why he should not be promoted to 11th dan.
Some people have elevated Kano to 11th dan, so that he is the highest rank.
Good teachers should have students who surpass them. There is no inherent reason why ranks should decrease over time rather than increase.
First, I just want to discuss the meaning of the term, "grandmaster", because I think some clarification is needed, since there's no universally accepted definition for it. Later I'll get to how styles promote new heads of their systems and what their ranks are after that.
Some styles do only have one "grandmaster" who holds the highest rank of the style and outranks all others. But more commonly I see the term being used by many people of the same style, and they may use "supreme grandmaster", instead, to refer to the one person above all others.
But again, this depends on the style. And it depends on how it is being translated to English.
In the Western world, "grandmaster" is often overused or abused by martial arts teachers as a way of marketing and getting more students, and therefore making more money. It can be a legitimate rank, but it's often self-awarded or even bought (from fake rank certification companies) after teachers form their own style or school and place themselves above all others as the grandmaster.
Heck, in every major city in the U.S., you'll find no shortage of "grandmasters". They're everywhere. That doesn't make any sense if they're supposed to be the absolute head of their styles, with nobody ranked higher than they.
In kung-fu arts, "grandmaster" is often just a way of referring to your master's master. Chinese martial arts commonly use familial relationships to refer to everyone's place within the style. Such as: "grandfather teacher" (sigung), "father teacher" (sifu), "mother teacher" (simu), "older brother" (sihing), etc. There are even terms for kung-fu aunts and uncles.
Japanese styles often use the term "soke" to refer to the head of a style (nobody outranks him/her). In English, we might refer to them as the grandmaster of that style. But that's just a translation. Soke really just means "head of the family (style)".
As for how succession works when the old head of the style dies or retires, that also depends on the style. Usually the old head of the style names his/her successor.
In many cases, however, the old head dies before having a chance to name a successor (or doesn't want to name a successor if he/she thinks nobody is worthy). Ideally when that happens, the most advanced ranked students vote on who will be the next head, and whomever is chosen becomes the highest ranked member of the style.
But this has rarely happened as ideally as one would hope. As has often been the case in the past, some of the most highly ranked students insist that someone else should be the new head. Or maybe they themselves might think they deserve to be the new head instead. When that happens, styles are split apart as splinter branches form, each laying claim to the head of the style. It can get bitter and messy.
And to answer your question about ranks decreasing over time: No, that isn't how it works. Instead, the new head of the style automatically gets promoted to the highest rank. In many Japanese and Korean styles, that might be 10th degree black belt. So if the old head of the style was 10th dan, and you're a 9th dan prior to his death, then you get automatically promoted to 10th dan when you become the new head of the style.
Some styles also have special ranks for the founder of their styles. In that case, nobody will ever again be as high in rank as they are. So if the newest head of the style is ranked 10th dan, then maybe the founder of the style is posthumously ranked 11th dan.
Hope that helps.