While some inequality is necessary for the progression of the society in the short term, it may becom problematic if it becomes permanently entrenched to the extent that people are incapable of changing classes on their own merits. For example: heirs to old money will have greater access to financial capital to set up their own entrepreneurial activities; cronyism and nepotism may occur at the apex of the corporate ladder (e.g. fathers with substantial holdings in a company have the power to hire their own children and friends onto the company board when equally qualified outsiders are locked out, the old-tie connection at private schools and universities), creating somewhat of a corporate feudal system. This needn't extend to corporations; governments are also prone to cronyism when plutocray takes hold, especially around election time (notice that the wealthiest are granted the biggest tax cuts). It should also be noted that not all citizens are capable of improving their social standing for many reasons, including non-fluency in the lingua franca, disability, gender, location, level of education, social connections, genetic qualities, existing accumulated family wealth.
In addition, inequality in the system has the potential to create some other dire social costs, ranging from crime, maintenance of the prison population, disparity in the health and education systems, suburban relocation (e.g. gated communities, inner-city slums), corruption, mass industrial activitiy, rioting. At the extreme end, you only need to look at the number of revlolutions in the past 300 years, including france, russia, cuba and other socialist coups d'état as examples of where the poorest people succeeded in overthrowing the wealthy elite of their respective societies.
The compulsion to kep up with the Jones' has also lead some people astray. It is well documented that work-related stress is a big contributor towards depression and the desire to consume and live like the upper classes has plunged many into bankruptcy through excessive debt.