Yes, all Bitly links are case sensitive. This means if your link is “bit.ly/CustomWord”, then “bit.ly/customword” will not work and bring you to an error page, or to another long URL destination, as it is registered as a different link.
You can make both work, however, if you edit your link to the other casing options. This is because when you edit a link, the older versions of that link will still work. So, if you need to, edit your all caps link to be all lowercase, or any variations you think your recipients might type in.
Editing your link combines metrics for all versions of the link, and they all will direct to the same initial long URL.