At the moment the number of tags allowed on a question is 5. Both client side and server side this limit is now1 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 ...
While this isn't possible with the API, it is possible with SEDE, if you allow for up to 7 days of lag.
Here is the SEDE Query that lists Tag wiki and excerpts in descending edit date order:
when w.lasteditdate > e.lasteditdate