The double game was actually quite dramatically exposed in Abbottabad with the OBL raid. Can the NSA explain why Pakistan did that in spite of claiming to be part of the US led alliance? Just one instance for example.
Their was no double-game in the matter of Osama Bin Laden.
Scores of militants had infiltrated Pakistani cities (including Abbottabad) to escape attention of authorities and/or to destabilize the country. Take a tour of
Durand Line and you will understand how militants manage to slip through at times.
Moreover, it takes time to find an individual who had (local) facilitators and also had cut himself off from technological wonders of the world in order to minimize the 'set of clues' that might lead relevant authorities to his doorstep one fine day. It is like trying to identity a needle in the haystack.
ISI - alone - could not achieve a breakthrough in search for OBL due to a number of factors. It cannot account for corruption within the state and neither it can make-up for ineptness of police. Additionally, ISI was preoccupied with state-wide crackdown of terrorists and nabbing them and could not afford to invest a major chunk of its resources in search for a single terrorist. Sooner or later, ISI might have knocked on the doorstep of the raided compound as well. Who knows? Professionalism - alone - is not good enough.
CIA has global clout and enormous set of resources at its disposal, to do its bidding. It could afford to invest a huge chunk of resources in search for OBL besides other stuff.
ACR reveals that CIA and ISI had been actively cooperating in the hunt for OBL during the period 2001 - 2005. This cooperation ended when Obama administration came to power. For some reason, Barack Obama had a negative perception about Pakistan and Leon Panetta assured the administration that CIA has the capacity to get things done on its own.