Yes, it is possible, but it requires some agreement between both parties.
If the "sysadmin" (it does not need to be a sysadmin by the way) wants his own backups, does he want all the transaction logs and differentials in addition to a full backup?
If the "sysadmin" only needs full backups, he can do his own backups WITH COPY_ONLY and use them without disrupting the DBA's scheduled coverage of backups. This could be used for populating a test server or creating a test database.
However, the DBA team is normally guarding the recoverability of all the data on the server, which means that the DBA backups (FULL, DIFF, LOGS) could provide any point-in-time restore that the backups cover. That is why you do not want the other backups to interfere.