There are actually two stages of Han Dynasty in China.The Western Han (206 B.C.--8 A.D.) and the Eastern Han period (25 A.D.--220 A.D.)
The downfall of the Western Han dynasty was mainly due to emperors who neglected their duties.After the Han Xuan emperor died,there not a single emperor after him that was serious about the political and economic conditions of China.However,people only started to rebel after Wang Mang implemented new economic policies that increase the backwardness of the economic conditions in China instead of improving it like he initially thought.
The downfall of the Eastern Han empire was mainly due to political rivalries that occurs in the palaces between eunuchs,the family empress dowager,and the subjects.The emperor was nothing more than a mere puppet in the hands of these people.Though some emperor manage to gain political power in these endless political rivalries,he was unable to solve the problem faced by the people of China at that time such as drought and the invasion of foreign armies.The problem became worse after the Yellow Turban rebellion,where many warlords divides China after they crushed the rebellion.Some famous warlord at that time are Cao Cao,Yuan Shao,Liu Bei,Dong Zuo.Eventually these warlords fought together untill China was divided into three kingdom,Wei (Cao Cao's territory),Shu (Liu Bei's territory) and Wu (Sun Quan's territory).Of these three kingdom,only Liu Bei called his kingdom the Han dynasty,due to him having a common ancestor with the royal family.Wei manages to conquer all other kingdom (under the rule of their subject,Sima Zhao and his son,Sima Yan).Sima Yan overthrow the Wei kingdom and established the Jin kingdom.Han dynasty was no longer seen ruling China after that.Although some warlord call their kingdom Han after the collapsed of the Jin dynasty,they remained as warlords,conquered only a small quarter of China.