If you speak more than one language, can you speak them all in their respective accents as well?

For instance, English in (your regional dialectal accent). French, in a French accent, and so on. In other words, you sound like a native speaker in each language you speak. Oddly enough... I can speak Tamil in a perfect Indian accent (primarily because that IS my dominant accent), French in a perfect French accent (as pointed out by native speakers), and Russian in a perfect Russian accent (for phrases I do know well--as pointed out by native Russian speakers). YET... I fail to speak English without my Indian accent coming through, despite it being the only language I know fluently, my primary language, etc. There is definitely a generic American accent to my voice plus a slight southern drawl to it (I mean... Florida.) But I've never been able to speak English devoid of my Indian accent. I think I'm able to do French and Russian so well because they have more shared phonemes with Indian languages than with the English language, haha.