If you define it as 'a hard time recovering from blows', then we should also bear in mind that you could argue that the Imperial Chinese state (which I guess was what you were juxtaposing Rome against) also suffered from this particular malady. Tang China never recovered from the An Shi Rebellion (instead fragmenting into various local warlords the central government never surmounted for about 150 years); the Zhou didn't recover from their collapse at Hao (instead collapsing into various warring feudal principalities for around 300 years); neither did the Han after the Yellow Turbans (again, collapsing into semi-independent warring powers for the last 30 years of its life) or the Song after the initial Jin/Mongol victories (the Southern Song never recovering lands lost to the Jin or to the Mongols).
Careful with some of these examples.
The Tang did recover from An Lushan's rebellion, but it adopted a more decentralized form of rule. The Warlords only became a problem when the Tang suffered a second decline, and rather than a solidified enemy that could be defeated the Warlords were many pocket emperors. Even then the Tang wasn't completely killed off until the Northern Invasions by proto-Mongols and proto-Manchurians.
More important than the Yellow Turbans would be the wave of migrations into China. While it is mostly on the periphery of the Chinese records and popular culture, millions of people from the north, west, and east were moving into China. Liang province was wartorn from the Qiang rebellions. Armies on the frontiers were at near constant war with these groups, and warlords increasingly compromised with them as the central government fell from one military leader to another until the establishment of Cao Cao who more or less settled the issue until after his dynasty collapsed with the Jin. After almost a century of fighting, the population of China went from 50-80 million people to about 15 million. The Yellow Turbans were not the catalyst for this, merely a reflection of the larger problems the empire had when dealing with the outside problems along with various droughts and famines. That 15 million may have even been a rise from a low point during the Three Kingdoms.
Ultimately though, the Jin went into another devastating civil war in the so called war of the 8 princes, which left China in ruins and the demographics were in such a sorry state that foreigners outnumbered Chinese. The Sinicized Xiongnu descended from a Xiongnu Khan and a Han Dynasty Princess then decided to make their bid for power, claiming to restore the Han Dynasty under themselves. However, because of cultural differences and the way the Han court looked upon him and his family (especially the maternal link to the old dynasty), the name was changed to Zhao and another civil war was ushered categorized by ethnic conflict. However, because of Chinese bureaucracy, the strength of the educated rural scholar gentry, and the dependence of a series of foreign dynasties on the Chinese themselves, the new groups became Chinese and the character of the conflict became one of Civil War rather than a foreign one. There was nothing systemic at any of these stages that would have prevented a recovery, simply that a confluence of greater factors caused them to continually domino until one last dynastic squabble and the rejection of a foreign dynasty as legitimate by the genty precipitated in two centuries of conflict.
The Song Dynasty actually forged an alliance with the Mongols and succeeded in crushing the Jin through a pincer movement. Mongols come from the North, the Song from the south. The two might even have kept the alliance for a bit, until the Song behaved in its xenophobic "I'm the ruler of the universe and you are secondary barbarians at best, and I deserve the land of the Jin that we agreed to split" that the two went to war as the Mongols were, in a way, even worse than the Song when it comes to tolerating a power not submitting to them. Nevertheless the Song state was powerful enough to continually resist the Mongols for the better part of half a century, and only truly met its end when the the Song dynasty became a headless state. The Mongols even almost entertained having a semi-permanent peace with the Song several times, only to be rebuffed. When Kublai began to see himself as the new Emperor of China in the Chinese sense the Song was doomed, if only because their own domestic guns could not protect them when the bureaucracy was increasingly fatalistic and thought burying the emperor's head in the sand and fighting over who gets to be empress and the next person to have his head buried in the sand took precedence. I would say that was the failure of the Southern Song, more even than its inability to accept an equal power dynamic with the Mongols - the court sought to above the problem by doing nothing about it. As I said, it took almost 50 years. Entire generations grew up with the Mongol threat on the horizon, and people trusted that the status quo would remain - or saw the end as inevitable.
However, I believe this is something of missing the point. While a new Dynasty in China is something of a new state, at the same time it was seen as the same country. The dyansty is what embodied the naming institution, but the undercurrent was more or less the same (with natural evolution with time). Rather than say that few Chinese Dynasties recovered, I would put forward that each new Dynasty was a recovery in and of itself.