"I hope so commander, for your sake. The Emperor is not as forgiving as I am."
When watching RotJ, I laugh a little every time I hear Vader saying those words. I remember the scene in Esb, when the Imperial fleet comes out of hyperspace, and General Veers informs Vader that they have been detected and that a force field is activated on the planet, because Admiral Ozzel thought that a surprise attack would be best. And Vader's response:
Quote: "Admiral Ozzel is as clumsy as he is stupid"
Moments later, the recently diseased admiral is replaced with Piett. Now, the result of Ozzel's actions was that the Empire had to launch a ground assault, costing two AT-ATs and an unknown number of troopers, as well as allowing most of the rebels to escape, including Luke Skywalker, the main reason Darth Vader attacked in the first place.

Killing people when they fail is not forgiveness, but Ozzel's failure did mess up Vader's plans, and naturally made him very angry. Ozzel made a very big mistake. Vader did not, however, kill Piett when the Millennium Falcon escaped into hyperspace. Piett was responsible for disabling it, yet an astro droid reengaged it in mere seconds. The technicians should have made that impossible by removing essential parts, but they didn't, so Piett failed. Still, Vader didn't kill him, after all, Vader did plant a seed of doubt in Luke's mind. So Vader is capable of forgiveness.

If Palpatine is less forgiving than Vader, how far can you go before ending up electrocuted? Would he kill a servant for making his tea the wrong way? "Ouch! It's too hot! You failed me for the last time!" There are no scenes in the movie where anyone fails Palpatine (the ground forces in RotJ did fail, but Palpatine didn't live to punish them), so we don't really know how forgiving he really is, only that he's less forgiving than Vader. I hope we see Palpatine accepting apologies in the live action TV-series (any updates, btw?).