One example of a fully loaded tag set is a question about the House and Senate rules regarding the impeachment of a US president that could use eight tags. These would include:
Note that having used five relevant tags, there is no room for the last set of tags. The absence of those tags does not prevent such a question from being found, since a search using impeachment, president, united-states will locate the question.
There are 31 30 questions tagged house-rules or senate-rules that do not include any of house-of-representatives, senate or congress. (Two tags were also excluded to remove questions unrelated to the US.)
I did tag (and other) edits on six questions. Four to add relevant tags, one with no effect on tags, and one to remove a tag that reduced the count. In each case and IMO, the questions are well-tagged without tags for the House, Senate or Congress. (Though some other tags are questionable and I ignored minor edits.)
12 questions use both house-rules and house-of-representatives
71 questions use both senate-rules and senate (Two are missing united-states.)
29 questions use both house-of-representatives and congress and not senate
44 questions use both senate and congress and not house-of-representatives (Five are missing united-states.)
Seven of the above questions are missing united-states. None of those had five tags.
While I did not review those 156 questions, some containing seemingly redundant tags did reach the five tag limit.
The is no search type for the number of tags. Trying to find a good case for not having redundant tags would require a scattershot approach; nonetheless keeping redundancy to a minimum may be beneficial.
I have posted a question: How can I limit search results to questions with exactly five tags?, on Meta StackExchange to see it there is a way to do this.