80+ bronze units (most group regulated designs.) are pretty bad at this. Sub 20% load can see less than 80% efficiency so the draw from the wall can be pretty bad. From 20-50% the "hill" on the load chart takes awhile to climb. Then, once you cross ~75% or so it starts to drop off again. What does this mean? Lets look at some examples. You build a PC but because your budget is low you can't spend a lot on the PSU. You get one of those $35 or $40 PSUs, might even say Corsair, Seasonic, etc on the side. It's a quality PSU, but because of the money spent it has some drawbacks. There are two loads I worry about with my PC. Idle/surfing/streaming, and gaming. Idle for most modern systems is ~60W, with surfing or streaming around 100W. Gaming depends on the parts used, but lets assume you have a gaming system with a 300W load. A bronze 500W PSU while idle or streaming will barely hit 20% load. As mentioned above, efficiency can be less than 80% in these cases. Not always! But it's possible and typical. This means at best, you are getting 82% efficiency. If the load being asked for is 100W, this means you'll be drawing at least 122W from the wall. When gaming you are close to the 50% load figure so you'll be just under the best efficiency and you'll be drawing from the wall ~357W for that 300W needed.
Now lets pretend for some reason when you bought your PSU for some reason you could get a 750W 80+ bronze. At idle or lower draw loads you won't even be 20% of the output. 100W/750W = 13.33% load. At this low load of the PSU, you won't even get the 82% bronze offers. Lets assume 75W. This means instead of drawing 122W, you are drawing 133.33W. Gaming will probably be a similar draw. So the answer is yes, maybe. It is possible to put in a bigger PSU and have it draw more power. In other cases it might not matter.
What does matter more is buying something better overall. Gold has not only better efficiency numbers, but a flatter curve as well. A 500W at 100W idle is at it's 20% load, so instead of getting 82% you get 87%. Instead of drawing 122W you'll only be drawing 115. Gaming should be around 337W draw. That's 20W less. Gold units are finally priced well and it's possible to get the money back from a lower power bill.