tl;dr
Assuming everything is set-up properly (e.g. plural form expression), I’d imagine you have multiple instances of the same text in your files, but only some of them are translated, or the domain does not match.

Long answer:

Consider these calls:
[php]
<?php
__( ‘One’, ‘domain’ );
_e( ‘One’, ‘domain’ );
_x( ‘One’, ‘context’, ‘domain’ );
__( ‘Many’, ‘domain’ );
_e( ‘Many’, ‘domain’ );
_x( ‘Many’, ‘context’, ‘domain’ );
_n( ‘One’, ‘Many’, 2, ‘domain’ );
?>
[/php]

They produce the following output inside the .po(t) file (generated by the latest version of Poedit at the time of writing).

[code]
#: functions.php:2 functions.php:3 functions.php:8
msgid "One"
msgid_plural "Many"
msgstr[0] ""
msgstr[1] ""

#: functions.php:4
msgctxt "context"
msgid "One"
msgstr ""

#: functions.php:5 functions.php:6
msgid "Many"
msgstr ""

#: functions.php:7
msgctxt "context"
msgid "Many"
msgstr ""
[/code]

If you follow the msgid and msgid_plural strings, you’ll make some observations. For example, you have 2 occurrences of “One” but 3 occurrences of “Many”, although in the PHP code above, each word appears exactly four times.
Obviously, the calls with a context create a new entry in the .pot file to differentiate from the calls without a context.
What is really interesting however, is that the first block that contains the _n() strings is grouped together with the first __() and _e() calls of lines 2-3, but there is separate block generated for the respective calls of “Many” in lines 5-6.
So, if there is any chance you (or the translator) skipped the translation of msgid_plural “Many” what exists in the .po file is probably this:
[code]
#: functions.php:2 functions.php:3 functions.php:8
msgid "One"
msgid_plural "Many"
msgstr[0] "Translated One"
msgstr[1] ""

…

#: functions.php:5 functions.php:6
msgid "Many"
msgstr "Translated Many"

…
[/code]
Now, if you called:
[php]_n( ‘One’, ‘Many’, 1, ‘domain’ );[/php]
would return Translated One

But if you called:
[php]_n( ‘One’, ‘Many’, 2, ‘domain’ );[/php]
would return Many

Assuming “Many” has been translated in the third block:
[php]
// Returns ‘Many’
_n( ‘One’, ‘Many’, 2, ‘domain’ );

// _n() returns ‘Many’.
// __() gets the value ‘Many’ as its first parameter, finds the third block (which is a match), and returns ‘Translated Many’
__( _n( ‘One’, ‘Many’, 2, ‘domain’ ), ‘domain’ );
[/php]
would seem to return the proper Translated Many, but as far as the whole gettext/WordPress thing is concerned, it’s a completely different string.

Hope this helps :)

Also, as for the `_nx()` you probably know this by now, but just for the sake of it, [the documentation](https://codex.wordpress.org/Function_Reference/_nx) is pretty clear to me and confirms what you said there.

Well, this depends on how you theme is built and how it’s translated (e.g. through plugins, .po/.mo files, etc).
Whoever set the site up for you, will be able to help.