The Constitution of India does not allow holding Indian citizenship and a citizenship of a foreign country simultaneously including OCI card holders.
Overseas Citizenship of India (OCI) Scheme was introduced for persons of indian origin by amending the Citizenship Act, 1955 in August 2005.  The Overseas Citizenship of India (OCI) is an immigration status permitting a foreign citizen of Indian origin to live and work in the Republic of India indefinitely. OCI is an immigration status, not a citizenship status.
The OCI was introduced in response to demands for dual citizenship by the Indian diaspora, particularly in developed countries such as North America,.
The Indian embassy in New york has clarified that holding an OCI card in no way entitles its holders to claim the status of dual citizenship. OCI holders cannot become dual citizens.
The OCI documents consist of OCI Registration Booklet and a Universal visa sticker. It is mandatory for registered OCIs to carry their passports which carry the Universal visa sticker for entry into / exit from India. All Persons of Indian origin (PIO) card holders with valid PIO cards since 2015, had to convert their PIO card to OCI card.
A registered Overseas Citizen (Non Resident Indians)Â of India is granted multiple entry, multi purpose, life-long visa for visiting India, he/she is exempted from registration with Foreign Regional Registration Office for any length of stay in India.
OCI costs: US $ 275 or equivalent in local currency for each  applicant. In case of application submitted in India, the fee is Rs.15,000/-
Renunciation
Indian can also renounce citizenship at missions abroad. A person ceases to be an overseas citizen of India under subsection (1), every minor child of that person registered as an overseas citizen of India, shall thereupon cease to be an overseas citizen of India.
A stateless person cannot apply for an OCI, but there is a risk of renouncing indian citizenship. An OCI holder can be considered stateless if they lose citizenship of the other country.
The only way for indians to legally become foreign national is to renounce indian citizenship.
The MEA website has a special page  here
