A friend gave me the following reply in terms of dynamic hedging and portfolio management:
Pricing models for a CB are based on holding the CB hedged with a short equity position. The combined portfolio has zero delta. However, it has positive gamma. To see this, consider that delta increases when the probability of conversion increases. Of course, this means that it increases when spot of the underlying equity increases.
Now, a dynamically delta-hedged portfolio should be gamma-neutral except in the case of jumps. In a CB pricing model based on diffusion and a jump to default, an increase in the probability of a jump (which would happen if the CDS spreads go up) means an increased chance of making profit from being long gamma. In the event of a jump, our short equity hedge will pay off much more than the CB will lose.
This explanation, while correct, is problematic for someone who isn't used to arguments based on dynamic hedging. Like the question about why we would want to hold a hedged vanilla (which has negative theta, and so steadily loses money even if nothing happens!), it seems that it should be possible for a different hedging strategy to justify a different valuation. Of course, any other portfolio will not be delta-neutral, and so will preserve some risks which are not present here. However, I tried to come up with a different way of answering this question which is accessible without understanding dynamic delta hedging.
The price of a CB goes up if credit risk goes up ceteris paribus - all other things being equal. In particular if the equity price remains the same but the CDS spread goes up, the CB will go up.
If the credit risk of a companay has gone up, but the equity price is unchanged, then the potential upside of the equity must have increased by enough to compensate the increased downside risk (from the point of view of the holder of the equity). But the CB is exposed to most of the upside of an equity (because if the equity rises enough the CB will be converted), but only a fraction of the downside (because in a credit event, the equity is likely to drop to zero, before any of the recovery value of the CB is impaired).
So if the increased upside and increased downside are balanced for the equity, the upside should more than compensate the downside for the convertible bond.
On the other hand, if the credit risk of the issuer increases in isolation of any other changes to its outlook, the CDS spread should rise and the equity price should fall. The combined effect on the price of the CB should be that it falls.