Click-throughs not being recoreded
Not all URLs get magically converted to being click-through tracking ones. Most internal ones, for example, do not (e.g. a link to a donation page with the cid and hash).

If you look at the email being generated (look at the actual html of the email, using something like 'show original'), you should be able to see which ones should be click-through ones because their path looks like this:

/modules/civicrm/extern/url.php

Here's a related inverse question:
avoiding tracking of some links

If your links really are click-through tracking ones and people are clicking on them, they'll get counted (unless the click-through links are badly formed due to an incorrect configuration, but that seems unlikely if you're not noticing a lot of other errors).

Update

If you go into your email client and copy a message you recieved from CiviCRM and then paste that as a new message into CiviMail, you will be copying the magically-generated click-through tracking urls from that past mailout (and ones that are specific to you). The new mailout will generate click-through tracking for the previous mailout, all as if it was you that clicked on the links. This is a super bad idea, don't do it.

In a better future world, CiviCRM should give you a big red warning if you try to directly include urls of the form /modules/civicrm/extern/url.php in your message source.