If you do one-off sites for mom and pop clients, then probably not. However you have a defined business model that either focuses on large windfall projects or residual income or _____________ then, yes it is possible.
It's not so much that 'web design' makes you money, it's mostly how you run and structure your business. You want to make sure your business model is scalable to the millionaire level and that it's even possible for you to get there.
Also diversifying revenue streams helps a bunch. If all your revenue comes from web design and all of a sudden the web design business drops off (for whatever reason) you're essentially out of business. However if your revenue stream was diversified then one dropping off may hurt your business but you would still be in business.