Possible cause of the one time login link sent via e-mail to be broken?
There must be something, Drupal does not create broken links by default. Sadly, we can't debug your site for you. literally are not able to. See here - URL is accessible as [user:one-time-login-url] token, so if nothing overrides default support for this token, it must be OK.
Inspect http://example.com/admin/config/people/account, Password recovery tab. By default it should look like this:

[user:name],
A request to reset the password for your account has been made at [site:name].
You may now log in by clicking this link or copying and pasting it to your browser:
[user:one-time-login-url]
This link can only be used once to log in and will lead you to a page where you can set your password. It expires after one day and nothing will happen if it's not used.
--  [site:name] team

If this token is alone in it's line, just like above, then you know something interferes with tokens. If it's not, then maybe some automatic HTML syntax correction kicked in, and messed up your layout? It is impossible to tell without actually debugging your site.