The way flashplugin-nonfree works right now is that it's a fairly empty package that downloads Adobe's Flash installation tarball, untars it, and copies the .so to the necessary directory. There's one more step in the middle though. It does a checksum on the tarball after it downloads and a checksum on the .so. These checksums are to make sure you're getting the right file, as opposed to some trojaned version (say, if Adobe was cracked). These checksums are also a bit of a problem.
See, every time Adobe updates Flash, they keep the same file name. So the package downloads the file, and since it's a new version, the checksum fails, causing the install to fail. A few days later, after some testing, MOTU uploads a new package with the updated checksum.
You can avoid this.
Instead of installing flashplugin-nonfree, enable the Canonical Partner Repository in System -> Administration -> Software Sources -> Third Party Software, or make sure you have
deb http://archive.canonical.com/ubuntu intrepid partner(substitute in hardy or gutsy if you're using those) in your /etc/apt/sources.list and after
sudo apt-get update && sudo apt-get install adobe-flashplugin, you'll have a working Flash plugin, even if there's been an update. The reason is that Adobe's package includes the full binary. It doesn't try to download from their website, so there's no checksum to fail.
It looks like going forward, this will be the preferred Flash package, especially once Adobe releases the final 64-bit version.
If you want to switch from flashplugin-nonfree to adobe-flashplugin, make sure you purge flashplugin-nonfree first.