It may not be inherently at odds, no, but I haven't seen any evidence or studies that social mobility is in any way increasing - and it does seem counter-intuitive that social mobility could increase at the same time as income inequality.

http://www.economist.com/node/15908469

"How rising inequality affects social mobility is still unclear. Those born since inequality started to rise sharply are only just now becoming adults. However there are some troubling signs according to two papers to be presented at the Tobin Project, an alliance of scholars, this month. Christopher Jencks of Harvard University finds that income inequality has been accompanied by a widening gap in college attendance. Ms Sawhill argues that a rising correlation between income levels, likelihood of marriage and level of education will make society more stagnant."

But high income inequality and low social mobility (low social mobility isn't particularly in question) together as facts are at odds with a significant likelihood of folks to lift themselves out of poverty. Proving the "ability" is an existence proof about the future probability of anyone in the class escaping poverty being nonzero, and sets a very low bar for what is acceptable for society, in my opinion.