Does killing a former accent count? I grew up in the PacNorthWest and mostly Alaska, but my dad was from Missouri, so obviously growing up with him I had some slight Midwestern accent stuff going on in my way of speaking. Once I reached permanent workaday world age (20ish), I started eradicating sayings and ways of saying certain words.
Now I have an “Alaskan accent” such as it is. 
What’s awesome is being in a roomful of people who can provide a colorful accent on demand – there is no one ‘British accent’ but my students will cheerfully render posh rhetoric into whatever local patois is requested.
So, thanks, student from Dorset who gave a terrific ‘Friends, Romans, countrymen’ speech in West Country PigBen Josh for me, and a nod of gratitude to the student from Ukraine who ‘acted out’ a couple letters from Pliny the Younger.
I’m teaching a module on Greek comedy in the spring and damned right we’ll be acting out scenes from Lysistrata with accents a go go.
I can soften my native accent to something vaguely Estuary, but many times any more just talk in my hacksaw-through-cheap-veneer Delawarean accent. When my students go for American accents, though, I usually say, ‘Right, um…don’t. Just, don’t.’
My question regarding the bolded parts is: convincing to whom? If you’re fooling Joe Random-American-or-at-Least-Not-English, that’s not the same as fooling someone who is English and has (one of the many) English accent(s).
A follow up question is: how do y’all who’ve said their fake accent is convincing actually know it is? If your proof is that no one calls you out on your fake accent or asks about it dubiously … Hate to break it to you, but that ain’t proof. It’s quite possible to notice a bullshit fake accent but not say anything or let on you know because that [del]could[/del]would be a bit aggressive and dickish.
I’ve never affected an accent, but I do tend to pick up the accent of the person I’m talking to pretty much unconsciously; I guess I’m sensitive to such things because of my linguistic and theatrical experience.
I remember talking once to an Irish guy (a friend of a friend) in a pub and unconsciously parroting his accent. I think he was pretty POed until he realized I wasn’t doing it deliberately.
I once did Posh English (think Patrick Stewart) for a Shakespeare production in front of audiences filled with Brits. Their reaction when they heard me speak normally was invariably shock: “What, you’re American?!?”