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: 
Ultimately, in order to answer this, you need to check your cardholder's agreement, and/or call your issuing bank and ask them. Card behaviors around credit balances and purchases larger than your limit will vary from bank to bank.

Most banks will be happy to let you carry a credit balance (after all, that's essentially you lending them cash for free), but by regulation they are required to make a good-faith effort to refund your money within 6 months. Few banks will take action before that limit without you prompting them to (i.e. explicitly asking for a refund). However, it sounds like you plan on implementing this "immediately" before making your transaction.

You may run into issues with this, as some issuers implement a rule that says that no single transaction can be equal to or larger than the credit limit on the card - even if you have a credit balance, you can never charge more than the limit. However, other banks are totally happy allowing that to happen if you have a credit balance large enough to keep the balance under the limit.

So - call your bank and ask. Or, ask the merchant if you can split the transaction into several smaller ones. Or, call your bank and ask for a higher credit limit.
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.