There's a few layers to this question. The first thing I have to point out is that the title of "Sheriff of Nottingham" did not exist during the reign of Richard I and John, but it's one of those things that changes over the retelling of a story. The activities of the character in the Robin Hood stories make it pretty clear that this was not a city sheriff anyway, but a county sheriff. The actual title was "Sheriff of Nottinghamshire" and we have a complete list of names for the men who held it.
The men who held the title in the reigns of Richard I:
These men were unremarkable as sheriffs. With the exception of William de Ferrers, who fell ill and died a few weeks after his appointment, they seem to have just quietly done their jobs. None seem to have done much wrong, and none were removed for being especially bad at the job. Roger de Lacy, for example, was a highly competent nobleman who gave up the position to join Richard I on the Third Crusade. Then things deteriorated.
Brewer was a judge, and had been Sheriff of Devon before the Third Crusade and one of the men who ran England while Richard was away. When the crusade was finished and Richard I returned to his lands, he asked for a reshuffle of official positions. Brewer was appointed as Sheriff of Nottinghamshire in this reshuffle. But our sources agree that he wasn't good at it. He was an opportunist and seriously corrupt. But he was also deeply loyal to the crown and exactly the sort of guy John wanted in his inner circle. When John became king in 1199, he moved Brewer around various counties as their Sheriff but had to keep moving him because he was so unpopular. Records from the time show that when he was Sheriff of Cornwall, the people of Cornwall raised money to bribe John into removing him. The same thing happened when he was appointed Sheriff of Somerset, and then again when he was Sheriff of Wiltshire. He sucked so much that three counties bribed the king into removing him. Brewer's career speaks to the desperate political situation John found himself in if Brewer's loyalty was worth being this much of a mess.
Bardulf was a respected judge, and had been a steady hand both as a judge in court cases and in a senior role in the Exchequer (the government body responsible for taxation), which he held for decades. However, by 1199 he was unwell and died by 1203. It is likely he retired due to his deteriorating health. Robert was a career administrator, and one of the best men in John's government. Like Brewer, Robert was moved around a lot, often in Brewer's wake. It does seem that, at times, Robert was the man sent in to clear up Brewer's mess once it had become too much to deal with. He was later appointed as the sheriffs of Devon and Wiltshire for this purpose.
These two men were so awful that clause 50 of Magna Carta calls for their removal by name. However, the 1215 version of Magna Carta was cancelled, and the watered down 1217 version no longer called for Garard's removal. Complaints about him seem to have focussed on his closeness with John and his status as both a mercenary captain (a group often looked down on by the nobility) and as a foreigner (he was French). In the last weeks of his life, John appointed him as Sheriff of Lincolnshire as well as of Nottinghamshire, though he was also paired with Nicola de Haye. This was highly unusual as being Sheriff was usually not a joint office, and it was even rarer to appoint a woman to anything, let alone such a high office. Nicola was a powerful noblewoman, with a proven track record in both the administration of justice and command of military forces, so maybe it wasn't that far fetched. It may have helped that William Marshal (who briefly ruled England after John's death) had less of a problem with women in charge than many of his contemporaries and had way too much on his plate to properly deal with the low quality of English sheriffs in John's administration. The partnership between Philip and Nicola seems to have been a disaster, with their names rarely appearing together in documents which suggests that, in fact, they did not work together at all and instead saw each other as rivals. In the end both were removed from their posts in 1217 when William Marshal died and the new government did a wide ranging reshuffle of officials.
So, actually, we know quite a lot about the people who held the title (or at least the title that was actually held by the character in the Robin Hood stories). Some were good, some were not, some are quite revealing about the precarious political situation in the reign of King John.