Question: I made some changes to a CSS file, uploaded it and saw no change. I cleared my browser's cache, repeated the process and still nothing. I also tried another browser and then experimented with other files - all the same result.I then deleted the CSS file altogether - the website still looks the same and I can still see the files in the browser's console. 

I can only get results if I actually change the file names altogether (which is really inconvenient). I dont think there is an issue with FTP overwriting the file as there are no errors in FileZillas logs. 

Is there another way a website can cache its self? Would anyone know why this is occurring?

EDIT:

I also tried this in cPanel's File Manager and viewed it on another PC - same result

Answer: 
[Squid](http://en.wikipedia.org/wiki/Squid_%28software%29) and other web accelerators often sit between a hosted server and your browser. Although they are supposed to invalidate their caches when the backing file changes, that information isn't always sent to specification or acted on properly. 

Indeed, there can be multiple caches between you and the server each of which has a chance of hanging onto old data.
Browsers can cache things, did you try SHIFT-F5 on your webpage to force a reload of everything?