In order to enter a directory, it needs to have executable permissions. A setting of 444 would mean only read permissions. Hence, it is completely normal that you would get a "permission denied".
Code:
win32sux@candystore:~$ mkdir /tmp/example
win32sux@candystore:~$ chmod 444 /tmp/example
win32sux@candystore:~$ cd /tmp/example
bash: cd: /tmp/example: Permission denied
win32sux@candystore:~$ chmod u+x /tmp/example
win32sux@candystore:~$ cd /tmp/example
win32sux@candystore:/tmp/example$