Can a competing cryptocurrency make Bitcoin obsolete? If a new cryptocurrency is created, with some unknown property that Bitcoin does not have, can Bitcoin be adapted to include this property, or will Bitcoin become obsolete?

(I believe I know the answer, but I still think it's an important question. Spurred by this reddit post)
If it is a feature for the client that doesn't affect the Bitcoin network (like say, stronger wallet encryption or something like that), it is not a problem.

If the feature is related to the cryptographic security (say, SHA-2 becomes somehow obsolete), then adoption might be problematic. First of all, the developers would have to prepare the change and schedule it in advance (for example, "Starting with block X, we will be using SHA-3"). This would require all the clients, both standard and alternative, to get updated before that time or risk not being able to connect to the network.

Moreover, if the change is related to a vulnerability of a cryptographic algorithm of Bitcoins, then the entire network might get compromised before the update could happen. For example, if securing the old coins would require one to send them to a new address, then in a short period of time before everyone would do it, someone could potentially steal a lot of coins. Luckily, a scenario like this is very unlikely.

All in all, adopting new features into Bitcoin is possible, although time and consideration is required. Bitcoin's value is in its stable nature, so revolutionary changes could only destabilize it.