QuickBooks Error 5503 may be categorized as – QuickBooks data has been uploaded, but did not hook up to the server. Error Code 3005, 5502 will also be the exact same category of errors.

Real cause of QB Error 5503?
This QuickBooks error occurs due to a challenge with Intuit online services or third-party developer application connections to QuickBooks. The data in your QuickBooks company file was uploaded successfully, but an error occurred while downloading the changes created by applications using Intuit online services or third-party developer application connections to QuickBooks.

How to fix the QB Error 5503?
Troubleshooting QuickBooks Error 5503

Note: Please make sure before you perform the specific problem solving steps for the error you are facing, you need to ensure Sync Manager is up to date. You can ignore this error, in the event that you don’t use any online application that modifies your computer data.

Intuit servers could be busy or unavailable at that particular time. It is possible to retry the Sync after sometime.
If the above solution does not resolve the issue or answr fully your question, you can read discussions and post messages and questions associated with your issue on the Intuit QuickBooks Community site introduced by Intuit or perhaps you can interact with the QuickBooks tech support team for guidance and error fixation.