Twitter is one of the worst culprits. Currently there is no way to search for any tweet more than a few days old.

Google's realtime search used to provide the ability to search and retrieve tweets from a specific date and time in the past, but twitter cancelled that deal and haven't provided any decent replacement.

A publicly accessible archive has huge potential for research. A friend of mine used Google's realtime search to pin point and keep record of the first tweets out of Christchurch following the Christchurch earthquake. Sadly there is no way to do this now and twitter don't seem to care.