"I think that the covers severly hinder your playing style (you cant pluck near the bridge, or above the neck pickup), but i know that James Jammerson would rest his hand on top of the pickup cover, and just use his index finger to play right above it."
Well, the neck pickup cover was a staple on my Jazz 4 banger when I had it. It somehow sounded very different.
As far as I can guess, the metal covers would have affected the shape of the magnetic field the pickup created. It sounded more focused. I might n ot want that now, but I'd put 'em on again.
I didn't like the bridge cover on because that hampered my fingerstyle by the bridge, but the neck cover didn't affect my playing enough, and in fact led me to playing up more on the neck instead of resting my thumb on the neck pickup.
If I hadn't been broken out of that I'd still be thinking that that's the 'proper' way to play bass. I'd heard when learning to play that you're 'supposed to' rest your thumb on the pickup.
I'd suggest playing with the neck pickup covered if any, and keeping your ashtray for resale or use as an ashtray.