SPOILERS:
Harry, Ron and Hermione spend weeks moving from place to place, failing either to destroy the Locket or to think of a lead to find the others. Bickering amongst each other in frustration rises due to the Horcrux’s poisonous influence.
Eventually they stumble across other fugitives – a party of Muggle-born wizards and goblins on the run. They learn that the Sword of Gryffindor has been substituted with a fake copy.
They question the portrait of Phineas Nigellus (which Hermione hid in her bewitched bag) and learn that the sword should be capable of destroying Horcruxes, as it's impregnated with Basilisk venom.
Ron, his self-control obviously affected by the Locket around his neck, loses his temper with the group's constant failures and leaves after bitter words were hurled at one another in a fight. He suspected Harry and Hermione were more than just friends, which wasn't true. Hermione was saddened by this event.