Question: Some mates and I have the amount to cover the cost of an item but have a credit limit lower then the price. The purchase has a hard requirement that the payment method is a credit card. So, my mates and I put a negative balance on the card (I.E: The balance is 0 and then we pay something like 16,000 on it, making the balance -16,000 so we can make the large purchase putting the -16,000 balance back to 0).

Could we make the purchase using this method?

Answer: 
As others have said, it depends a lot on the card issuer (and country), and by experience, it can vary with time as well.

I used to put a very large credit balance on one of the Amex cards I had, in order to have the ability to pay amounts much larger than the credit limit on that card (I wanted to use that specific card for some miles-related reason I don't remember).

This worked very well for months, until one day when I was at the other end of the world they called me to tell me they couldn't accept that any longer, and they immediately refunded the credit. Which was quite annoying as I suddenly couldn't pay some large bill exceeding the limit with that card.

As far as I understand it, Amex have made explicit changes to their T&Cs for at least some cards to make it clear they don't want you do to that (they may even reject payments that would make your balance go into credit).

Your best option is to see if you can pay the amount through other means. Most merchants would actually much, much prefer a bank transfer over a credit card payment any day of the week for such an amount: credit card fees for $16000 are usually several hundred dollars. Also, a credit card payment is much more likely to get a "chargeback" (cancellation/dispute) than a bank transfer.

Your next best option is to call the card issuer to ask whether they can either increase your limit or accept an overpayment **AND** that overpayment would actually add to your total limit.
Probably not. It may depend on your credit card, but I know for sure my credit card doesn't allow this. If you have a credit limit of $10,000, your credit card may not allow you to make a purchase greater than $10,000, even if you've prepaid several thousand in advance.

What you may be able to do instead is to pay for the item over several days. That worked for me.