With a NEET score of 463 and belonging to the BC (Backward Classes) category, you might have a chance of securing a seat under the government quota in a private medical college, although it depends on the cut-off scores for that specific year and the category-wise allocation of seats. Government quotas in private medical colleges often have varying cut-offs based on state and institutional policies. It's advisable to check the previous years' cut-offs for private medical colleges in your state, and participate actively in the counseling process to explore available options.