late '13 Mac Pro w/ 10.11.3 cannot enter sleep mode

Please disconnect all USB devices and test.

Please test after taking each of the following steps that you haven’t already tried:

Step 1

Take all the applicable steps suggested in this support article, including the ones under the heading "If the previous steps don't work." That's the starting point for any further effort to solve the problem.

Note that, as stated in the article, the computer will not sleep if some sharing services are enabled. From the menu bar, select

 ▹ System Preferences... ▹ Sharing

Turn off all sharing services.

Also note that the "external devices" referred to in the article could include Bluetooth devices, other than those made by Apple. If you're using a MacBook, MacBook Air, or MacBook Pro, or if you're using a desktop Mac with a wired keyboard and mouse, turn Bluetooth off to test.

Step 2

If iTunes Music Sharing or Home Sharing is enabled, disable it or quit iTunes. Do the same with any third-party applications that have a similar feature.

Step 3

Select

 ▹ System Preferences... ▹ Accessibility ▹ Speakable Items: Off

Step 4

Select

 ▹ System Preferences... ▹ Bluetooth ▹ Advanced...

and uncheck both boxes marked

Open Bluetooth Setup Assistant at startup if...

Step 5

Reset the printing system.

Step 6

Back up all data.

Triple-click the line below on this page to select it, then copy the text to the Clipboard by pressing the key combination command-C:

/var/vm

In the Finder, select

Go ▹ Go to Folder...

from the menu bar and paste into the box that opens by pressing command-V. You may not see what you pasted because a line break is included. Press return.

A folder named "vm" should open. Inside it, there may be a file named "sleepimage". Move that file, and only that one, to the Trash, but don't empty yet. You'll be prompted for your password. Close the folder window.

Restart the computer and empty the Trash.

Please test after taking each of the following steps that you haven't already tried. Back up all data before you begin.

Step 1

Quit the Contacts or Address Book application if it's running.

Triple-click anywhere in the line of text below on this page to select it:

~/Library/Caches/com.apple.AddressBookSourceSync

Right-click or control-click the highlighted line and select

Services ▹ Reveal in Finder (or just Reveal)

from the contextual menu.* A folder should open with an item selected. Move the selected item to the Trash.

Repeat with each of these lines:

~/Library/Containers/com.apple.AddressBook ~/Library/Containers/com.apple.AddressBook.ContactsAccountsService

*If you don't see the contextual menu item, copy the selected text to the Clipboard by pressing the key combination command-C. In the Finder, select

Go ▹ Go to Folder...

from the menu bar and paste into the box that opens (command-V). You may not see what you pasted because a line break is included. Press return.

Step 2

If you use iCloud, uncheck the box marked Contacts in the iCloud preference pane. Press return to confirm, or click Keep on Mac in the dialog that opens.

If you synchronize contacts with any other network service, disable that synchronization in the Internet Accounts preference pane.

Step 3

This step applies only if you can launch Contacts. From the list of groups on the left side of the Contacts window, select All On My Mac. If that group doesn't exist or is empty, skip this step. Otherwise, from the menu bar, select

File ▹ Export... ▹ Contacts Archive...

Save the exported archive to the Desktop.

Step 4

Quit Contacts if it's running. Reveal the following item in the Finder as you did in Step 1:

~/Library/Application Support/AddressBook

If you know when the problem started, and you can restore that folder from an earlier backup without losing any contacts, do so and test. Otherwise, move the folder to the Desktop, leaving the window open for now.

Step 5

Launch Contacts. Your contacts and groups will be gone. If you took Step 3, then double-click the file you created in that step and confirm that you want to import from it. Your contacts and groups will reappear.

Step 6

If you took Step 2, reverse it. Test. If the issue is resolved, delete the AddressBook folder and the archive file on the Desktop and close the open folder window.

Otherwise, quit Contacts again and put the AddressBook folder back where it was, overwriting the newer one that will have been created in its place.

Step 7

Go back to Step 3, but this time choose Export vCard... from the Export menu. Continue as in Steps 4-6. All contacts should be preserved, but you may lose some metadata such as group membership.

Although there has been no announcement, according to reports, Apple is replacing defective D500 and D700 video cards under a warranty extension program. If you have either of those cards, see below.

Make a "Genius" appointment at an Apple Store, or go to another authorized service provider. You may have to leave the machine there for several days.

Back up all data on the internal drive(s) before you hand over your computer to anyone. There are ways to back up a computer that isn't fully functional—ask if you need guidance.

If privacy is a concern, erase the data partition(s) with the option to write zeros* (do this only if you have at least two complete, independent backups, and you know how to restore to an empty drive from any of them.) Don’t erase the recovery partition, if present.

Keeping your confidential data secure during hardware repair

Apple also recommends that you deauthorize a device in the iTunes Store before having it serviced.

*An SSD doesn't need to be zeroed.

I don't know. I suggest you take the machine in for testing anyway. If nothing else, you can find out whether the problem is caused by your display.