When trying to print multiple SSRS reports simultaneously there is a default maximum of 20 per user.

 

If there is an error   Request failed with Http status 503 This means that number has been exceeded.

 

In order to change that default, do the following:

 

Open the file

C:\Program Files\Microsoft SQL Server\MSRS13.SQL2016\Reporting Services\ReportServer\rsreportserver.config

 

The highlighted part of the path name will change with the SQL version. Change the following key and restart the reporting service.
 <Add Key="MaxActiveReqForOneUser" Value="20"/>