In broad strokes, it's largely inaccurate. While early Chinese historians may have taken steps to emphasize orderly succession and legitimation, the real picture was much more complicated.
Let's begin with the pre-imperial era - that is, the so-called 三代, the three earliest "dynasties": The Xia, Yin/Shang, and Zhou. The traditional histories, including Sima Qian's influential Shi ji present them as successors of each other. The last Xia king (Jie) was a tyrant, so the Shang overthrew him... The last Shang king was a tyrant, so the Zhou overthrew him... and so on.
As the Shi ji's 殷本紀 Annals of Yin present:
At this time Jie of Xia was oppressive, and his rule dissipated, and one of the princes Kunwu rebelled, so Tang, levying an army, put himself at the head of the princes. Yiyin followed Tang, who, grasping a halberd, marched against Kunwu, and then attacked Jie. Tang said: Come, ye multitudes of the people, listen ye all to my words. It is not I, the little child, who dare to raise a rebellion. The ruler of Xia has committed many crimes. I have indeed heard the words of you all, but the Xia ruler is an offender, and, as I fear the Supreme god, I dare not but punish him. Now, as the Xia ruler has committed many crimes, Heaven has charged me to destroy him..."
A similar proclamation from the Zhou against the Shang is presented in 大誥, The Great Announcement, found in the Shang shu.
But there's a problem here: The Shang records do not mention the Xia at all, and in fact, their very existence is in dispute (answer I wrote earlier about them). Both the Shi ji and the Shang shu were also written centuries after the events they purported to record. Furthermore, from an archaeological point of view, dating these dynasties are extremely problematic, and material cultures that they're identified with co-exist (as in, it's not a linear replacement). KC Chang in Shang Civilization admits that multiple conflicting accounts of the dynasty’s demise probably existed as early as the Eastern Zhou, but nevertheless continues to generally accept the textual evidence. Chang ultimately had to reconcile the archeological data with the textual record, so he did not accept that the Xia, Shang, and Zhou succeeded each other in a neat order per se, but rather argued they were “subcultures of a common – ancient Chinese culture” and more akin to warring political groups. He famously called them "interaction spheres" and his argument is still pretty influential in mainland Sinology, though less so in the West.
So yeah, the Three (Pre-Imperial) Dynasties did not succeed each other in a linear way at all.
Let's turn to the early imperial era, which begins with the First Emperor's Qin state absorbing the other six major states of the last Zhou period. Since that "unification" there have been multiple "periods of disunion" - time periods when there were multiple competing dynasties, each arguing they were the legitimate one.
Rulers of the ill-named "Three Kingdoms" period, for example, each claimed to be an emperor. Cao Pi (also termed Emperor Wen of Wei) forced the Han emperor to abdicate in favor of him (See Records of the Three Kingdoms, "Annals of Emperor Wen" 14) and proclaimed that the Han's fortunes had fallen and they had lost the Mandate of Heaven. But Liu Bei, ruling in Shu, obviously disputed this and saw his state as a continuation of the Han Dynasty, and that the imperial Liu family had not lost the Mandate. (see J Michael Farmer's "What's in a Name? On the Appellative "Shu" in Early Medieval Chinese Historiography") While Wei came to be seen as the legitimate successor dynasty to the Han, its status as such was really only because the Jin state which supplanted Wei was able to unify China.
Similar issues arise when examining periods like the Northern and Southern Dynasties - the Northern and Southern states often labeled each other as "false/pretender" 偽. While the Southern Dynasties were considered by many historians to be the legitimate Chinese dynasties (hence the problematic term "Six Dynasties", the Sui state that unified the country was northern. Thus succession was hardly linear here either.
... Of course, I haven't even discussed whether or not the Mandate of Heaven really ever existed, but that's a question for another time and person, and hopefully this helps.