unless you're using wordpress as a cms for a informational database and your search function is limited to the tags on the posts (yes, that is a real scenario i had) i would suggest no more than 4 or 5 tags per post, and make sure that no category page has the same posts as the tag pages, and utilize custom excerpts (not the auto generated ones) and doing that you can dramatically inflate page count, and increase indexed pages on the site as well
i have a series of 11 sites that are all searchable technical databases, and yes, the posts have sometimes upwards of 20 tags on them, and yes, the tag pages are indexed (average indexed pages for each of those sites is over 700), and yes, depending on the search anywhere from 3-7 of those sites show up on page 1 for the targeted keywords, and they target a total of over 70 keywords in the same niche. only wikipedia, amazon and a fortune 50 company outrank them (some searches there are other fortune 100 companies i haven't pushed off page 1)
based on my experience with that
and the fact that those sites haven't had a thing done to them for seo since 2010
tag pages if used right can be a godsend
Again, I stress the fact that having tag pages indexable is good and useful but ONLY if you pay a lot of attetion to the type of listing pages they produce. For most people this is a lot of hassle so it is better to just noindex them.
he said it right there
also, like i said in earlier post
utilize custom excerpts, make sure you don't have any two tags with the same content, make sure you aren't obviously keyword spamming, and if you do this...
DO NOT USE THE TAGS WIDGET
instead, have the tags show up at the bottom of the post