When trying to Save, the screen quickly refreshes, but appears as if you did not click the Save button. No visible errors are presented. However, if the record is unable to Save, this is an indication of a problem.
Example A: Saving Appointment record without access to Time Blocks object
Specifically, saving appointments has produced this error when a User does not have Read access to the Time Block object. Since Time Blocks are embedded in the appointment Edit page, trying to Save an Appointment without being able to see Time Blocks will cause this issue. The easy fix in this situation is to give that profile read access for time blocks.
Step 1: Run a debug log on your user, then attempt the Save action again. Check the the debug log to see if the system can tell you anything more about that attempted save operation. To capture the debug log...
Go to Setup > Debug Logs > click View next to the log you wish to open
Click Download to download and inspect the XML
Step 2: Note any Object Lookup relationship fields in the Edit screen. Check the User Profile to be sure the User has Read access to all Objects/Lookup Fields in the Edit screen. In this example, the User needs Read access to the Time Blocks object, in order to successfully Save an Appointment record. To change a User's Profile settings...
Go to Setup > Users > Select the User who is unable to Save
Click on their User Profile > click Edit
Scroll down to the Custom Object Permissions settings
Look for the Time Block object, be sure the User has at least Read capabilities
Save changes to the Profile