Just changing the file upload limit in WordPress multisite may not give you the results you desire. Instead it becomes necessary to change the file size limits on your server as well as WordPress can only handle files as big as what your server can handle. Fortunately this is a very easy process. Note that some of the syntax below may change based on your Linux distribution the lines to change in php.ini will be the same regardless.
sudo nano /etc/php5/cgi/php.ini
Next find and edit the following two lines to your desired values:
post_max_size = 8M
upload_max_filesize = 20M
In both cases the M stands for megabytes. For example if you want to allow 1 gig file uploads you could, for demonstrations sake make each value 1024M.
Save the file and restart apache. In Ubuntu use the following command
sudo /etc/init.d/apache2 restart
Next, go to your network admin settings and change the Max upload file size box accordingly. Note that while the settings in your php.ini can be larger than your WordPress settings, you cannot set your WordPress settings to a value larger than those in your php.ini.