I get this problem today, when trying to upload a mp3 file using WordPress Media Library. Upload process stopped before it even started. What happen?

PHP config limit my upload size, that’s why WordPress cannot upload file to its folder. PHP asked WordPress to stop the process. 

How to solve this problem? it’s very easy
Open PHP configuration file, php.ini on any text editor,
look up for Maximum allowed size for uploaded files.

e.g upload_max_filesize = 2M -> change 2M to whatever size you want

But I must warned you, not all server have same way to edit PHP Configuration, i suggest you to asks your server admin how to change it