Linux
You could append text to the file even if you can't read it. Check this example:
Create a file with one line
nine@nine-laptop:~$ echo "Some text" >/tmp/testfile.txt
Show that file
nine@nine-laptop:~$ cat /tmp/testfile.txt
Some text
Make it write only
nine@nine-laptop:~$ chmod 200 /tmp/testfile.txt
Show file with permissions
nine@nine-laptop:~$ ls -l /tmp/testfile.txt
--w------- 1 nine nine 10 nov 14 10:54 /tmp/testfile.txt
Try to read it (which won't work)
nine@nine-laptop:~$ cat /tmp/testfile.txt
cat: /tmp/testfile.txt: Access denied
Append text to end of file
nine@nine-laptop:~$ echo "Appended text" >>/tmp/testfile.txt
Set permissions to read and write
nine@nine-laptop:~$ chmod 600 /tmp/testfile.txt
Show the file (which now has one more line)
nine@nine-laptop:~$ cat /tmp/testfile.txt
Some text
Appended text
Windows
Adding weeks comment: It works the same in Windows. Exept there are several rights in write permission, like "Write data", "Append data", "Write Attributes", "Write Extended Attributes"..