Maybe it’s my boorish americanism showing but it seems like british people don’t do a very good job of faking an American accent - Hugh Laurie excluded.
I’d say it’s about the same both ways. There are plenty of bad attempts at American accents, and there are plenty of appalling attempts at British accents.
The best ones are the ones where you don’t realise the person isn’t the nationality in question. There’s plenty of those, too.
It depends on who’s doing the listening.
Should we also exclude Jamie Bamber (Apollo from BSG)? Or Marian Keyes (Chuck from Pushing Daisies)? And as for the accents of the Bionic Woman and Sarah Connor, well…
Toni Collette does a passable American accent.
Anna Paquin does a passable Southern American accent. Stephen Moyer does an odd Southern American accent, though he says that he deliberately makes it odd because he’s an 150 year old vampire.
I think British is easier because there are so many dialects to choose from. I can do a passable British accent. I think the hardest is the Australian accent. The guy who does the voiceover for Outback Steakhouses sure sounds phony.
Near as I can tell, an authentic Australian accent sounds fake in places. It just doesn’t all seem to fit together in the same dialect.
Conventional wisdom among American actors (at least the ones I’ve met) is that British actors do better with American accents than vice versa. Supposedly it’s easeir to flatten out your vowels than make them more round, but maybe they’re just making excuses. In reality DSeid is probably right. It’s easy to find fault with someone else when he’s doing your accent.
You didn’t think Jimmy McNultyin the Wire had a good American accent?
I think the problem is that many people don’t realize how many British people there are working with American accents. Generally speaking when Americans (actors or otherwise) do any kind of British accent they are horrible at it. It’s a running joke amongst expats I know about how bad an accent is going to be when a stranger comes up to you in a bar telling you they can do the best British accent ever. Many of them genuinely believe they can, because we find it amusing to agree with them that they have a great talent.
Then there’s the fucking Irish Spring people: “And weeee like it toooo.” Irish fucking Spring can’t find any actors with real Irish accents?
(I’ve been wanting to get that off my chest for YEARS!)
Or Stringer? Or… well, there were a ton of British actors on that show.
That’s the flip side of the point I was just making: it’s easy to criticize someone doing your accent, and probably it’s just as easy to overrate your own skill at doing other accents.
This. I doubt one American in ten knows that Minnie Driver, Anna Friel, Joely Richardson or Louise Lombard are British. Hell, I didn’t know Louise Lombard was British until I happen to be on a CSI-related Wikipedia trawl.
ETA: I don’t like to credit Minnie Driver, though, just for her fucking awful Russian accent in Goldeneye.
I moved this thread to IMHO (with a stopover in Cafe Society) from Great Debates, but if most of the examples are going to be actors, Cafe seems like the best fit. Moved again.
British actors on the BBC are absolutely appalling at American accents. Yanks ain’t much better, but I think we have the edge on crapness.
Bob Hoskins in his role in Who Framed Roger Rabbit really had this American fooled.
One problem I hear sometimes, like when listening to BBC Radio 4 podcasts like “The Now Show,” is that there’s a bit of a twang in an American accent that wouldn’t otherwise have one. Might be from too many cowboy films, maybe too many clips of Bush talking. GWB, I mean - GHWB would produce a Mr. Rogers-style voice. 
(The host of the kids’ TV show Mr. Rogers’ Neighborhood.)
Who’s that British guy on the TV show in the US? The show about mind readers or something? That guy does a good job of a US accent.
I watch TV a lot. Can you tell?
I’d guess that British actors are more likely to have been exposed reasonably early to authentic American accents through the media than vice versa; and also that they have more incentive to learn to do a good one.
That reminds of a great gag in “The Wire” where he had to go undercover as a john. He decided to pose as an English businessman and was doing this really horrible British accent saying shit like “Blimey” and “Crikey” - the joke being of course that Dominic West really is British.
In my admittedly limited experience, the really bad American accents are in British productions (radio, TV, audiobook) aimed at British audiences, by actors who presumably usually do British characters.
On the whole, I think most British/Irish/Aussie actors who work steadily in the USA do an excellent job carrying off their accents. And often, the foreign actors do REGIONAL American accents better than American actors do. That is, a Brit often does a more convincing Texas accent than an actor from New Jersey or California.
On the other hand, a LOT depends on how badly the actor wants/needs to perfect his accent and on who the audience is he’s trying to fool.
What I mean is, when the audience is largely American, it becomes VERY important for a British actor playing an American to get the accent right. On the other hand, if an American is playing a Brit on an American TV show, his accent can be highly flawed (maybe even laughable to a genuine Englishman watching) but still succeed, as long as the American viewers believe he’s English.
On the other hand, English actors playing Americans on BBC sitcoms or in West End plays often do HORRIBLE accents… but as long as they sound American enough to English viewers, no one will object. But if an American actor is playing an English role in a West End theater, I’d expect him to spend a LOT of time working to perfect his accent.
Of course, both England and the USA have many regions with very different accents, and an accent that works perfectly in one context might not work at all in another. British audiences are probably a bit more attuned to the subtle differences in regional British accents than Americans are to American regional accents. For example, Hugh Laurie sounds perfectly “American” to me. On the other hand, if you asked me, “What part of the U.S. is Dr. House supposed to be from,” I’d draw a blank. House sounds American to me, but I have no idea, based on his accent, WHERE he’d be from. For that matter, there are dozens of regional “Southern” accents in reality, but the differences are subtle enough to most Americans that an actor can affect the same drawl whether he’s playing a cowboy from West Texas, a Virginia aristocrat or a Mississippi redneck, and only the locals would gripe “THAT’S not how we talk here!”
Probably the worst American accent I ever heard was by Emma Lloyd in the movie “In Country” with Bruce Willis. The whole time, I was thinking that the character was adopted by the Kentucky family during a trip to England.