How to tell what's failing at making backups?
You can capture backup events using SQL Server Audit.

USE master;
GO

CREATE SERVER AUDIT ServerAudit
  TO FILE (FILEPATH = 'C:\temp\', MAXSIZE = 1 GB)
  WITH (ON_FAILURE = CONTINUE);
GO

ALTER SERVER AUDIT ServerAudit
  WITH (STATE = ON);
GO

CREATE SERVER AUDIT SPECIFICATION BackuAudit
  FOR SERVER AUDIT ServerAudit
  ADD (BACKUP_RESTORE_GROUP);
GO

ALTER SERVER AUDIT SPECIFICATION BackuAudit
  WITH (STATE = ON);
GO


Then you can view the Audit Logs from within SSMS (Security > Audits).

If you can't use audit (e.g. due to edition restrictions), you can use a server-side trace:

declare @rc int, @TraceID int, @maxfilesize bigint = 5; 

exec sp_trace_create @TraceID output, 0, N'c:\temp\backup_trc', @maxfilesize, NULL;

exec sp_trace_setevent @TraceID, 115, 1,  1;
exec sp_trace_setevent @TraceID, 115, 9,  1;
exec sp_trace_setevent @TraceID, 115, 3,  1;
exec sp_trace_setevent @TraceID, 115, 4,  1;
exec sp_trace_setevent @TraceID, 115, 5,  1;
exec sp_trace_setevent @TraceID, 115, 6,  1;
exec sp_trace_setevent @TraceID, 115, 7,  1;
exec sp_trace_setevent @TraceID, 115, 8,  1;
exec sp_trace_setevent @TraceID, 115, 10, 1;
exec sp_trace_setevent @TraceID, 115, 11, 1;
exec sp_trace_setevent @TraceID, 115, 12, 1;
exec sp_trace_setevent @TraceID, 115, 14, 1;
exec sp_trace_setevent @TraceID, 115, 21, 1;
exec sp_trace_setevent @TraceID, 115, 23, 1;
exec sp_trace_setevent @TraceID, 115, 26, 1;
exec sp_trace_setevent @TraceID, 115, 28, 1;
exec sp_trace_setevent @TraceID, 115, 29, 1;
exec sp_trace_setevent @TraceID, 115, 34, 1;
exec sp_trace_setevent @TraceID, 115, 35, 1;
exec sp_trace_setevent @TraceID, 115, 37, 1;
exec sp_trace_setevent @TraceID, 115, 40, 1;
exec sp_trace_setevent @TraceID, 115, 41, 1;
exec sp_trace_setevent @TraceID, 115, 49, 1;
exec sp_trace_setevent @TraceID, 115, 50, 1;
exec sp_trace_setevent @TraceID, 115, 51, 1;
exec sp_trace_setevent @TraceID, 115, 60, 1;
exec sp_trace_setevent @TraceID, 115, 64, 1;

exec sp_trace_setstatus @TraceID, 1;

select TraceID = @TraceID;


I've probably captured more columns there than you need...