Yes, you can, but it will take some work (if you had a SuperDuper! backup/clone, it would be much, much easier) What you'll need to do is the following:
1. Make sure you have version 10.10.5 of the "Install OS X Yosemite" file, both inside your applications folder, and "outside" of it.
2. Copy that file to someplace on your external drive, not in the same "folder/location" as the Time Machine Backup.
3. Assuming you have a USB Flash Drive that is at least 8 gig, you'll need to create a bootable OS 10.10.5 Install Drive. This link explains how to do that:
http://www.macworld.com/article/236...otable-os-x-10-10-yosemite-install-drive.html
4. Once that drive is created, boot your Mac from that drive. Then, use Disk Utility from that USB Flash Drive to Erase, Format, and if necessary (depending on your needs), Partition the internal drive.
4. Once that is done, do a complete, clean installation of OS 10.10.5 onto your internal drive. Then, boot your Mac from your internal drive.
5. Finally, use Migration Assistant (inside the Utilities folder) to "migrate" (copy) all your non-Apple stuff from your Time Machine backup to your internal drive. You should then be back in business.
I did download OS 10.11 (El Capitan) last week but did not install it on either of my Macs. The reasons are:
1. A couple of my
critical third party applications have not been updated yet for El Capitan.
2. I "accidently" discovered issues with Office 2011 and OS 10.11, and especially Outlook (which I depend upon for my EMial).
3. An update to OS 10.11, OS 10.11.1, has been out form about a month, and is still in beta testing.
4. I have read further reports of issues with El Capitan. (I do not use Safari).