They did have the option to be independent, but that would mean they would have to provide for themselves when it came to defence, finance, infrastructure and so on. These were things that they already depended on the Indian Government for. So all of the states saw no use for independence and chose either India or Pakistan.

http://upload.wikimedia.org/wikipedia/en/3/3e/Partition-of-India-Spate-Jan-1948.jpg

Look at that map of the partition from 1947. The states that aren't highlighted/shaded are ones that haven't decided between independence or joining either of the two new countries.

"States declared for independence or of doubtful status..."

They were given the option to remain independent. Hyderabad actually decided to do just that. They did not join any side and declared their own independance. This was problematic for India and the people of Hyderabad since it was Landlocked by India. So initially in 1947 during Partition India and the Nation of Hyderabad agreed to a 1 year truce to sort things out. Hyderabad had a sizable army, but nothing compared to the Newly freed India. Over this year there were unconfirmed reports of Hindus(the majority in Hyderabad) being targeted by the Muslims and of a secret Pak-Hyderabadi alliance in the works. In 1948 as soon as the pact was up the Indian armed forces occupied Hyderabad, over through the Raja, and brought it into the Union.