In our case it ended up being a Kerberos issue. I followed the steps in this article to resolve the issue: https://techcommunity.microsoft.com/t5/SQL-Server-Support/Bulk-Insert-and-Kerberos/ba-p/317304.
It came down to configuring delegation on the machine account of the SQL Server where the BULK INSERT statement is running. The machine account needs to be able to delegate via the "cifs" service to the file server where the files are located. If you are using constrained delegation make sure to specify "Use any authenication protocol".
If DFS is involved you can execute the following Powershell command to get the name of the file server:
Get-DfsnFolderTarget -Path "\\dfsnamespace\share"