What is the max resolution output by the HDMI port of the MacBook Pro Early 2011
According to EveryMac.com, there were two 13-inch, Early 2011 MacBook Pro models—the 2.3 GHz i5 and the 2.7 GHz i7. It actually doesn't matter which you have. They both have a Intel HD Graphics 3000, which does indeed support an 2560 by 1600 external display. It's your adapter.