We've been seeing a lot of forum posts about this lately in relation to the Snapping toolbar, but it's strange that snapping stops working with classic snapping. Is it happening in the middle of an edit session where nothing has changed, or do you only notice it after reopening a map? In general, just make sure that classic snapping is still enabled, the snapping tolerance is reasonable, and that some layers and snap types are checked on in the Snapping Environment window.
We'd like to look into this further. If you or anyone else experiencing this with either the Snapping toolbar or classic snapping could contact Esri Support or your local distributor with your scenario and data, it will help us isolate the problem.
PS - The currently available ArcGIS 10 service pack is SP4 if you are able to upgrade to it, with SP5 due in summer.