I would think it would be likely they could.
We don't have the issue here but we have next to no wild rabbits here either. I think I've seen three on our property in 6 or 7 years.
Since the fur trade has been killed our predators are so overpopulated that small game has all but disappeared.
Of course our biggest issue is lose to predators but as for health I think they're as healthy or more so then ours that are contained. We don't feed them per say but they always have access to our horse hay barn.
Never seen any that look ill or in poor health and they breed like...well like rabbits.
We do rabbit round ups some times and sell some and at one time tried for months to catch them all. Got down to only seeing two around then like a month or two later we were back to like 20 then 30. It was crazy.