The problem has been solved! It was an error on my part as there was a piece of code that I had entered in functions.php which was causing the error.
As I said before the plugin is good and works well and Wesley has done a great job. Three cheers for him!
On the outset I want to state that I'm not sure that the problem has been caused by the wp-custom-admin-bar plugin.
What happened was that I created a website which had logins for the customers as well and I wanted to use wp-custom-admin-bar plugin to turn off the admin bar. Nothing fancy.
I tried the plugin on the local server trying to be a good guy who follows the rules. It worked fine there and the website is working well. Therefore I installed the plugin on the live server of my customer.
But as luck would have it. After activating the plugin I logged out and tried logging in. What happens is that after I give the correct username and password and click login I cannot see the next page. However, if I give the wrong login credentials an error is returned which tells me that the system is working well till this point of time. I have tried adding http://myseite.com/wp-admin/edit.php to try and see if the page loads or not but no luck.
The objective behind posting this incident is that I wanted to request the developer of the wp-custom-admin-bar plugin to share if any changes are made to the core login process of WP or not by the plugin and if they are can he please state those.
Thanks in advance.
PS: I would recommend the plugin though because it is one off case as I have installed it on two demo sites locally and it is working fine though the combination of plugins on the live website and the local website is different.