Due to their total dominance in middle school, all of them but their sixth man (Kuroko) became disillusioned with basketball to some extent and like many of those they defeated, they stopped believing the sport could be dominated through hard work and rather believed they themselves were unbeatable regardless of their teammates.
Kuroko was the member of the team who worked the hardest but the only one whose talent was to serve others on the court rather than himself. As a result, his primary goal in high school became teaming up with a young basketball team at Seirin High (and specifically partnering with Taiga Kagami in whom he sees a great potential) to defeat the Generation of Miracles and show them that his style of basketball is better than theirs so they will be able to enjoy the sport again.
At the height of their pride in their third year of middle school, three of them (Aomine, Murasakibara and Kise) stopped attending practices altogether and the only way they could motivate themselves in their games was to compete with each other over who could score the most points and in their third and final national championship together they won 111-11, with the final shot being them scoring on themselves to reveal they had just wanted to get the straight 1's in the score.
As Kuroko had been injured in the previous game by a player smashing him with a basketball out of anger (at the apathy of Kuroko's unbeatable teammates), he was unable to play at all. But the team they were facing included his childhood friend who he had spent the past three years promising to one day meet in the finals. Realizing they wouldn't even give his friend the satisfaction of playing seriously against him, despite captain Akashi promising they would, Kuroko was completely crushed.
Because the series starts in the first year of them entering high school, they are not generally all together at one time but they are spoken of consistently. Interestingly, despite being legends in the sport, even someone like Taiga Kagami (who specifically states he wishes to defeat them) met a number of them without knowing who they were, implying their legend extends beyond anyone actually seeing them play or the media that would have featured them in their prime.
Though all of the main six members were separated in high school, their manager (Momoi) went to the same school as the group's ace (Aomine) because they had been childhood friends and she is very similar to Kuroko in wanting to Aomine to lose so he can return to caring about basketball (though she also does her best to make sure they win because she knows a win/loss has to be genuine).
Kuroko's high school team does ultimately beat each one of the Generation of Miracles, though notably only one was not beaten in the Winter Cup (Midorima) as they tied each other instead in a qualifying match and Midorima was instead knocked out of the tournament by Akashi. However, Kuroko had beaten Midorima in the earlier qualifying matches for the Interhigh before then losing to Aomine, the only member of the Generation of Miracles who ever beat them in a game that year.
The only other match-up between Miracles in a tournament that first year of high school was Aomine beating Kise in the Interhigh. Aomine, Murasakibara and Akashi's teams all made it to the semi-finals of the Interhigh but none of the three actually showed up to their team's games so they never matched up with each other (resulting in all of them experiencing their first loss to Kuroko, rather than each other).
Kise was the first to lose to Kuroko (facing his team in a practice match at the very beginning of the series) but experienced his first official loss to Aomine and was the second last to officially lose to Kuroko. He was also the only member not to be needlessly cruel and antagonistic towards Kuroko before his defeat.
Notably, though the Miracles together were capable of beating other teams by a near 200-point difference in some cases, every single official game Seirin wins against one of the Miracles was won by a single point.