You can choose the types of attachments you want the Email Service to accept. Available options are None, Text, Binary, or All. The default selection for Accept Attachments is "None" and this needs to be changed to "All" within the Email Service settings.

 

Navigate to Setup > Develop > Email Services and click Edit next to Email to Case Premium (or whichever name you used during initial setup). Change the selection for Accept Attachments to "All" and verify the rest of the settings match the screenshot below and click save. Attachments will now be added as configured on the Inbound Configuration in Email to Case Premium as long as they don't exceed the Salesforce attachment limits. For more information on these limits, search Help & Training for "attachment limits".