Linux, Why can't I write even though I have group permissions?

Did you logout and log back in after making the group changes? See:
Super User answer involving touch permissions failure

Why can't Linux user edit files in group he is a part of?

I am using Ubuntu 12.04 and had the same problem where a user cannot write to a file to whom he is allowed group access to. For example:

whoami                                        //I am user el
  el                                            

touch /foobar/test_file                       //make a new file

sudo chown root:www-data /foobar/test_file    //User=root  group=www-data

sudo chmod 474 /foobar/test_file              //owner and others get only read, 
                                              //group gets rwx


sudo groupadd www-data                        //create group called www-data    

groups                                        //take a look at the groups and see
 www-data                                     //www-data exists.

groups el                                     //see that el is part of www-data
  el : www-data                               

Restart the terminal now to ensure the users and groups have taken effect. Login as el.

vi /foobar/test_file                          //try to edit the file.

Produces the Warning:

Warning: W10: Warning: Changing a readonly file"

What? I've done everything right why doesn't it work?

Answer:

Do a full reboot of the computer. Stopping the terminal isn't enough to fix these problems.

I think what happens is apache2 also uses the www-data group, so the task was somehow preventing the users and groups from being enforced correctly. Not only do you have to logout, but you have to stop and restart any services that use your group. If a reboot doesn't get it, you've got bigger problems.

I had the same issue, check if the folder has any more ACL rules or not!

If you can see + (plus sign) when you list folder, that means it has special access rules. For example:

[user_in_apache_group@web02 html]$ ls -l
total 16
drwxrwxr-x  16 apache apache 4096 Sep  4 13:46 ilias
drwxrwxr-x+ 15 apache apache 4096 Sep  4 13:46 ilias5

View the permission:

[user_in_apache_group@web02 html] getfacl ilias5
# file: ilias5
# owner: apache
# group: apache
user::rwx
user:user_in_apache_group:r-x
group::rwx
mask::rwx
other::r-x

So that means my user (user_in_apache_group) has no write permission for that folder.

The solution is what @techtonik said, add write permission for user:

[user_in_apache_group@web02 html]$ sudo setfacl -m u:user_in_apache_group:rwx ./ilias5

Check permission again:

[user_in_apache_group@web02 html] getfacl ilias5
...
user:user_in_apache_group:rwx
...

Hope it helps. ;)