A plug in outlet tester is an OK start, but it's time for the next step. Take an Voltage tester/ multi-meter with digital readout, and test an outlet that allegedly has an open ground. What is the voltage from the hot to the neutral? How about from the hot to the ground? Is there continuity from the ground slot on the front of the receptacle to a bare ground in the box. Is there high resistance from the bare ground to the neutral at the outlet? All of these steps will isolate the facts. You may have anything from worn out receptacles to a wiring issue at a junction box somewhere. Hopefully it's something simple.
What year was the house built, and what was it wired with? Are there any really hacked up renovations, or other electrical issues?