Bridge and hard money lenders price each deal, and their terms vary by lender and change often, so treat any percentage you read on a website as an example and get your own terms in writing before you underwrite. What stays true is that the money you can't borrow is the money that hurts here.
Take a $2M purchase with a $500k renovation. Even if a lender funds most of the purchase and most of the construction draws, you're bringing the down payment, the points and fees, and closing costs. Then the monthly carry: interest on a couple of million of debt, property taxes which vary enormously by state and can be several thousand a month on a house like this, insurance on a vacant property under renovation which costs more than an ordinary homeowners policy, utilities on a big house, landscaping, security, and staging once you list.
Call the all-in monthly carry a substantial five figures. Now the part that decides the outcome. The buyer pool at this level is thin, and thin means the house can sit. Underwrite four months to a sale and take twelve, and you need eight extra months of carry sitting in cash, plus whatever an extension on your loan costs.
So the realistic minimum isn't a down payment figure. It's the down payment plus fees plus a year of carry you could lose without going under. Most lenders on these products also want to see completed projects at similar scale before they'll fund one, so cash by itself doesn't get you through the door.