At the moment the number of tags allowed on a question is 5. Both client side and server side this limit is enforced at the time of posting.
There is no direct limitation to allow 6 or even 7 tags. As a matter of fact, this query shows 1038 questions at the time of posting that have 6 tags. The revision for one occurrence reveals what caused that 6th tag being added, a CM and their fancy scripts, bypassing all business rules.
However, allowing more tags might run into the length limitation of the field tags in the Posts table. That single field hold the concatenated tagnames, enclosed in < and >. It allows for 250 characters at the moment. As the tagname length has been increased to 35 characters, the technical limit is barely seven tags if those tags all have 35 characters.