You need to start updating things (Operating Systems, Hardware and Backup Software) as Microsoft is about to or possibly even has stopped support for Windows 2003
The support status for Backup Exec 2010 R3 is available here
https://support.symantec.com/en_US/backup-exec.15047.html
When you go to this site, look near the bottom of the right hand side bar and against the Select Version box, change it to 2010.R3.4 (this means SP4 for 2010 R3)
This lists 3 dates.
End of life date of March 5, 2012 (this is actually when we released BE 2012)
End of Engineering Life of September 5, 2014 (30 months after end of life date) - this means that from this date until EOSL date, no further in depth investigations into new fault conditions will be undertaken by technical support (as engineering will not be involved at all) . Technical Support will continue to offer advice or reasonable efforts answers based on existing documentation and knowledge only (so you can still speak to us and get assistance). We call this period partially supported.
End of Support Life/EOSL of Feb 1st 2017 (7 years from the release of the first 2010 version) this is when we stop accepting customer cases for this version.
So yes whilst you are still supported to run 2010 R3, it is only partial support as no further engineering action will be taken against this version. WIth the addition of Microsoft discontinuing support as well you may struggle to get full solutions to any future problems you may have.