It is definately possible to deposit a Foreign currency check in US. On most currencies like CAD or GBP it would be very simple and easy.
On INR as littleadv pointed out its up to the Bank.
Technically the Bank's can do it, however there needs to be enough volume to justify or support this. Typically the checks are sent over to India via a correspondant Bank, then once the check is encashed in India, the funds are converted and credited into your USD account. Typically the entire process takes around 4-5 weeks and has quite a few manual intervention steps.
You can encash USD checks in India, and they follow the same process of sending the check to US, getting the funds and then crediting the funds into your account. As the volume of USD checks getting encashed in India is high, most the Banks in India would offer the service. However if you try to encash say a Chinese Yuan or Indonesian Rupiya check in India, you will not find Indian Bankers encashing the same.
So the best way to get money moved from India to US would be to buy a Bankers check denominated in USD in India and take it along to US. Alternatively initiate a Wire/SWIFT from India to transfer funds to US account. Note most banks in India do not allow an online mechanism to Initiate a Wire out of India and one has to visit the Branch to get it done.