I imagine a compromise in Bitcoin's cryptographic foundation, such as SHA256, is what is being referred to here.
Addresses are protected by hashing a public key to the bitcoin address (https://en.bitcoin.it/wiki/Technical_background_of_version_1...). If you've always sent coins to new addresses like the hierarchical deterministic wallets do then your public key is only exposed after you spend the coins, so you are not susceptible to a cracking attempt on the public key.
This would be extremely interesting to watch as a new algorithm would render all (or at least most) existing miners useless as they use SHA256-specific-ASICs. The economic and control fundamentals of Bitcoin completely change if you change the hash algorithm.
Not to mention it takes non-zero time to choose a new hash, make the code changes, and deploy it worldwide. Even if somehow you do that in hours or days, what happens in the mean time? Bitcoin would be entirely unsafe to use and anyone using it as their primary financial store better hope they have some food stockpiled (and bills paid)!
Luckily I've not heard anyone question the math behind Bitcoin's proof-of-work algorithm, so I think this possibility is exceedingly unlikely as the math is well understood.
Bitcoin on the other hand would have to fundamentally change due to miners with ASICs. The code change might be minimal, but the economic, political, and psychological impact would be huge. For miners to lose their entire infrastructure investment in ASICs may cause an unrecoverable drop in hash rate as it takes 2 weeks for difficulty to recalculate.