I recall that in 2002, when Lula won the Brazilian presidential election, and was later inaugurated, most of the market participants assumed that Brazil would default on its sovereign debt, as Argentina did in December 2001. (I took the opposite view and did well for myself.) We used to quote CDS as par spreads back then - was was way over 6,000 bps (today we quote upfronts). Brazil sovereign hard-currency bonds were trading special of course. So many people wanted to short their benchmark bond (called "C bond") that its financing rate was 0! (But its yield was over 50%). You could borrow money for free if you were willing to be long it, while everyone else wanted to short it.
In this example, Brazil sovereign hard-currency bonds were trading special because many more people than usual wanted to short them - not because the credit spread was wide, not becase it widened, but because the presidential election was not expected (and misinterpreted) by the market.
But I think these days people are much less likely to short risky/distressed bonds than they were in 2002. It's easier to take the same view using CDS now. I'd expect even less connecton than there was 20 years ago.