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.
I'm not really clear why you don't just use a Visa Debit card for this purchase? I use Visa Debit for expensive purchases like long-haul international flights, hotels and health spas or whatever. It's really easy - I just transfer $12k or so into that account and then start spending. You can just put your $16k onto a Debit card and pay that way.

There are no monthly fees, and best of all, it is still backed by Visa just like a credit card. What I mean by that is, recently I was travelling around Europe and a dodgy vendor charged my card for 200 Euros that I didn't spend. The vendor didn't respond to my emails, so I complained to Visa and they reversed the payment. So you're just as safe using a Visa Debit as a credit card, and I never get stuck paying interest or late fees because I never owe Visa money. Debit cards are the best - I haven't owned a credit card in years.