Apple certainly does not require all files to go through iTunes or iCloud. There are many apps that allow file transfers to and from network attached storage. I've used Goodreader since it first came out. On first glance it is a PDF reader but it offers so much more and is a great file management app (although the UI takes a bit of getting used to). I connect to my NAS via WEBDAV both at home over local Wi-Fi and over the internet when away form home. Goodreader supports many other formats (AFP, FTP, etc.) and also allows you to get files from Dropbox, even your email - checkout the app store for details. Files can be pulled into Goodreader, then use Open in to open in a different app.
There are other apps that connect to WEBDAV (if your NAS supports this), such as Office2HD.
Interesting the OP mentions FileBrowser - I use that for watching movies directly off my NAS, for this purpose I prefer the interface to Goodreader.
As Walt asks, what are you trying to do specifically?