For advance tickets, no, it's not true. At the door, waiting in line, perhaps.

You're referring to the "suggested donation" of $20.00, which you might not have to pay if you wait in line.

I was a concierge for years and if there's an price for advance admission , there's a price.

As for The Met:
"Advance tickets to The Metropolitan Museum of Art allow the visitor to avoid waiting in admission lines at the Museum. Tickets include admission to the Met's collection and all special exhibitions. Tickets also include same-day admission to The Metropolitan Museum of Art and The Cloisters museum and gardens. Tickets are valid for up to one year after the date of purchase. "

However, this appeared on gothamist.com in 2006::

"Holy Temple of Dendur! The Metropolitan Museum of Art has raised its suggested admission fee to $20, up from $15. ***The important thing to note is that the Met does not require people pay an admission fee - you can donate a dollar or a few coins (or nothing at all) and you can still get in.*** While the Met says the increase was prompted by a $3 million annual operating deficit, we have to wonder if the Met raised its admission to compete with the Museum of Modern Art, which raised its required admission price to $20 when it reopened. " (Emphasis added .)

It goes on to say this:
"The suggested-donation policy is a requirement of being part of what is called the Cultural Institutions Group, a group of 34 New York City-owned institutions that also includes the American Museum of Natural History, the Brooklyn Academy of Music, the Brooklyn Museum, and the Bronx Zoo. As part of the same deal, the city provides 11% of the Met's total budget, according the Department of Cultural Affairs. In the last fiscal year, this came to about $24,598,000, an amount that contributed to general operating costs, as well as paying for heat, light, and power."

The only complete exception I've ever heard of was for Holocaust Survivors being able to get into The National Holocaust Museum in DC for free. I'm sure there are others.

You may want to call The Met and ask exactly how it works. I'm guessing, based on years of experience, that to pay less than the suggested donation, you're waiting on whatever line they have when you get there.