Applies only if your Time Machine is stored on a remote system, such as a Time Capsule, another Mac or some SMB server.

If you restore a different computer from you Backup, during the restore, the Time Machine volume becomes read-only (luckily enough). However, turning write access back on may fail.

This issue is difficult to debug.

First, attach the disk image and run a file system check:

admin@MyMac ~@ hdiutil attach -nomount /Volumes/Time\ Machine/MyMac_00112233.sparsebundle

Take note of the disk number, e.g. disk1, as we need it in the next command:

admin@MyMac ~@ fsck_hfs -f /dev/disk1   

If this is successful, you may now mount the image to check if that helped, but in my case it had not helped at all.

Here's the comparison with a fresh (empty) disk image created by Time Machine after I had removed the original out of the way. The original is Backup of MyMac and the fresh one is Backup of MyMac 1:

admin@MyMac ~$ ls -le /Volumes/Backup\ of\ MyMac*   
/Volumes/Backup of MyMac:
total 0
drwxr-xr-x+ 3 root  staff  102 Mar 10 14:09 Backups.backupdb
 0: group:everyone deny add_file,delete,add_subdirectory,delete_child,writeattr,writeextattr,chown
 
/Volumes/Backup of MyMac 1:
total 0
drwxr-xr-x+ 3 root  staff  102 Mar 15 15:16 Backups.backupdb
 0: group:everyone deny add_file,delete,add_subdirectory,delete_child,writeattr,writeextattr,chown
 
admin@MyMac ~$ mount |grep /Volumes/Backup 
/dev/disk1s2 on /Volumes/Backup of MyMac (hfs, local, nodev, nosuid, journaled, mounted by admin)
/dev/disk2s2 on /Volumes/Backup of MyMac 1 (hfs, local, nodev, nosuid, journaled, mounted by admin)
admin@MyMac ~$ diskutil list
[...]
/dev/disk1
   #:                       TYPE NAME                    SIZE       IDENTIFIER
   0:     Apple_partition_scheme                        *698.3 Gi   disk1
   1:        Apple_partition_map                         31.5 Ki    disk1s1
   2:                 Apple_HFSX Backup of MyMac          698.3 Gi   disk1s2
/dev/disk2
   #:                       TYPE NAME                    SIZE       IDENTIFIER
   0:     Apple_partition_scheme                        *698.3 Gi   disk2
   1:        Apple_partition_map                         31.5 Ki    disk2s1
   2:                 Apple_HFSX Backup of MyMac          698.3 Gi   disk2s2
 
admin@MyMac ~$ diskutil info /dev/disk1
   Device Identifier:        disk1
   Device Node:              /dev/disk1
   Part Of Whole:            disk1
   Device / Media Name:      Apple sparse bundle disk image Media
 
   Volume Name:              
   Mount Point:              
 
   Partition Type:           Apple_partition_scheme
   Bootable:                 Not bootable
   Media Type:               Generic
   Protocol:                 Disk Image
   SMART Status:             Not Supported
 
   Total Size:               698.3 Gi (749812383744 B) (1464477312 512-byte blocks)
   Free Space:               0.0 B (0 B) (0 512-byte blocks)
 
   Read Only:                No
   Ejectable:                Yes
   Whole:                    Yes
   Internal:                 No
   OS 9 Drivers:             No
   Low Level Format:         Not Supported
 
admin@MyMac ~$ diskutil info /dev/disk2
   Device Identifier:        disk2
   Device Node:              /dev/disk2
   Part Of Whole:            disk2
   Device / Media Name:      Apple sparse bundle disk image Media
 
   Volume Name:              
   Mount Point:              
 
   Partition Type:           Apple_partition_scheme
   Bootable:                 Not bootable
   Media Type:               Generic
   Protocol:                 Disk Image
   SMART Status:             Not Supported
 
   Total Size:               698.3 Gi (749812383744 B) (1464477312 512-byte blocks)
   Free Space:               0.0 B (0 B) (0 512-byte blocks)
 
   Read Only:                No
   Ejectable:                Yes
   Whole:                    Yes
   Internal:                 No
   OS 9 Drivers:             No
   Low Level Format:         Not Supported

So there obviously is zero difference, but still one is writable while the other is not. What worked in my case is to chown the directory in /Volumes to my username:

admin@MyMac ~$ sudo chown admin /Volumes/Backup\ of\ MyMac
admin@MyMac ~$ ls -le /Volumes/Backup\ of\ MyMac*
/Volumes/Backup of MyMac:
total 0
drwxr-xr-x+ 3 root  staff  102 Mar 10 14:09 Backups.backupdb
 0: group:everyone deny add_file,delete,add_subdirectory,delete_child,writeattr,writeextattr,chown
 
/Volumes/Backup of MyMac 1:
total 0
drwxr-xr-x+ 3 root  staff  102 Mar 15 15:16 Backups.backupdb
 0: group:everyone deny add_file,delete,add_subdirectory,delete_child,writeattr,writeextattr,chown

Again, nothing has changed, but from now on, the volume is writable