One thing that happens pretty often, is that you or your client got a hoster that’s doing quite well and serving things over secure connection, but you ain’t got a matching SSL certificate. In this case, WordPress will nag about it and refuse to update, or abort when you’re trying to do network updates.
This problem can be solved quite easily. WP is using its own HTTP API to make the connection, send the request and receive the data. Therefore everyone of those filters in the API is available to us. Here’s everything wrapped up in a nice plugin. Make sure to deactivate the plugin after you’ve updated.
Anyway, this is just a temporary work around. In the end, you should just fix your certificate and Ross McKay has a very well written tutorial for that. Thank you for the link in the comment, Ross.
Edit: There’s now a GitHub Gist that you can fork.