If you exceed your space allocation for a Google Site, instead of getting a nice sensible error message you will see ' Upload server error; please try again later ' when trying to upload attachments.
For a 'consumer' Google Site the space allocation is 100MB.
For a Google Apps Site (address starts sites.google.com/a/... ) the space for Sites is controlled at the domain level - you will need to contact your Google Apps domain admin to resolve this issue.
So if you see this message, the first thing to check - if you are using a consumer Site (i.e. not a Google Apps Site) - is whether you are close to the space allocation limit of your site. Go to More>Manage Site>General and look under Site Storage. If you are close to the limit, try clearing out some files (see above) and try again. Remember that deleting an attachment 'soft' deletes it and it will continue to occupy space for 30 days or until you 'hard' delete it by going to deleted items and selecting 'permanently delete. (Which could explain why some users have found this an intermittent problem - deleted items being removed after 30 days frees up space for the user to now be able to upload more files.
The current storage space limitations for Google Sites are detailed here.
For non-Google Apps users, the only way to increase the amount of storage allocated is to upgrade to Google Apps for which you need to have your own domain. You can purchase a domain via Google for (at time of writing) $8 / year. (see also move site to apps)
In either case it is possible to reduce the amount of storage used by a Google Site by being canny about where certain types of files are stored, especially picture files which can sometimes be large. This article discusses the use of Picasa Web albums to reduce the amount of Sites storage allocation used. Storing photographs and other pictures in a Picasa Web Album makes them easy to embed in Google sites pages and, importantly, does not eat in to the Site space limit. Additional storage space for Picasa/Google Docs is pretty cheap - at the time of writing it is $30/yr for 25GB.