When the selecting the Export as CSV option from either Report Builder or from a Dashboard view, if the report has more than 2000 records, you may encounter instances where the email containing the exported file does NOT deliver to the user’s mailbox. In this instance, the report failed to export entirely.
This occurs if the user who is completing the export DOES NOT have Read access permissions for ANY of the fields that are present on that particular report.
To check if this is the case, navigate to Setup > Manage users > Search for that user > Open the user and check the profile. Below is an example of how this screen should appear:

Next, navigate to Setup > Search for the object name on which the report is built > Select the fields > Click on View Field level security to check the access for these fields for this particular profile. Below is an example of how this should appear:

Resolution
Enable Read-Only and Visible access for the fields that do not already have this access on the profile.
- Navigate to Setup > Search for the object name on which the report is built > Select the fields > Click on View Field level security > Check the box Visible and Read-Only against that particular profile.
- The affected user will need to log out from the org and re-login.
- The affected user should attempt the Export to CSV again.
- The user should now receive the email with the file and data.