I feel the Roman Empire was a victim of its own success in so many ways, and had become unmanageable and bureaucratic. Clearly with any question like this there are going to be a number of different factors, but this to me is one of the root causes that allowed the other problems to become more troubling and eventually resulted in the downfall.
In such a large empire that lasted so long, there is no one answer that can be given. All of the people above pointed out different reasons of why the empire came to an end. And in my opinion all of them are good answers. However, I'd like to point out one new perspective. The eastern empire, that is, Constantinople did not fall. It lasted for much longer until the Ottoman Turks in the 15th century. And even during the 5th century much of the resources of the empire was sent to the east. The west had the glorious past reputation, and historians, like Ammianus, called Rome, the eternal city, but in reality, the real seat of the empire was in Constantinople. Therefore, only to point out the fall of Rome as the fall of the empire is to be too western in one's thinking. That said, if I had to take a guess on why the empire fell, it is due to military issues. Too many foes with too little resources spells the end.
The decline of the Roman Empire in the west was the result of economic annhilation. By the 5th century there were two realities, the west was suffocating from enormous taxation, severe government regulation, and what is known today as stagflation. The eastern empire was able to weather the storm because it economically (and to some extent politically) separated itself from its western counterpart. This suggests that the Roman Empire was separate in everything but name. The western empire suffered from stagflation, an economic situation where by the society suffers from increased unemployment combined with high inflation. The tax increases imposed on Rome failed to increase the needed revenue because the wealthy in Rome were able to defer and shelter their wealth from the tax collectors. This put the pressure on the 'middle class' to foot the bill. In essence, the empire folded because it destoyed the middle class by destoying their capacity to pay their taxes. I think there is a fair historical argument that the excesses of the Roman government (in the west) with its unrealistic tax mentality that led to stagflation which hindered business enterprise expedited in "The fall of Rome."
Edward Gibbons wrote seven volumes to answer this question. I really believe that many factors including an unworkable political situation, barbarian pressure from without, a weakening of values from within, as well as a series of severe economic crises all combined to cause the fall of the Roman Empire. In the final analysis, I would say that the Empire got too large to support itself in every way so that both internal and external pressures caused it to collapse. (There Edward Gibbons _ take that!)
There are so many possible factors to consider in this question! I agree with jillessa's response above, but I'd like to add another specific factor: political corruption.
As time went on, Rome's government became increasingly oppressive and authoritarian, and as a result it began to lose the support of the people. Officials were increasingly corrupt - for example, even the system of choosing an emperor became tainted. The Romans never successfully created a system for choosing new emperors. Originally, the choice was intended to be made through debate between the old emperor, the Senate, the emperor's private army (the Praetorian Guard), and the Roman army. However, the Praetorian Guard gradually gained complete authority to choose a new emperor. The new emperor would then reward the Guard, which became increasingly influential as time went on. In the last 100 years of its existence, the Roman Empire had 37 emperors - and 25 of them were assassinated.
The corruption of the Roman Empire's government divided the people of Rome at the very time when they needed to be most unified - when they were facing external attacks.