Most European cities have at least one major state-funded museum that opens its doors for free on a specific evening - usually the first Sunday of the month, or a Friday night window. The Louvre does it. The Uffizi has done it. The Prado has its own version. Rome’s civic museums have offered free entry on the last Sunday of each month for years. None of this is secret. It’s just not how most people build their itineraries.
The standard approach is to plan sights first and logistics second - book the accommodation, figure out how many days you have, then try to cram in the highlights. The free-entry windows get overlooked because they require reversing that order. You find out when the free slots are, then you build the trip around them.
This sounds fussier than it is. Say you’re spending a week in Rome. The Capitoline Museums are free on the last Sunday of every month. If your trip straddles that date, you simply make sure you’re in Rome that Sunday rather than, say, on a day trip to Tivoli. Entry to the Capitoline otherwise runs around €15 per adult. For two people, that’s €30 saved without any sacrifice - you were going anyway.
The Uffizi situation is worth knowing specifically. Florence’s booking system and pricing have changed several times in recent years, so always verify directly with the museum’s official site before assuming free entry applies. That caveat applies broadly: these policies change, sometimes seasonally, sometimes with little notice. The habit to build is checking the official museum site as part of your accommodation search, not after.

Where the Real Savings Stack
One free evening won’t transform a budget. But across a two-week trip through several cities - Rome, Naples, Florence, perhaps Madrid - hitting even three or four free slots adds up to €60–€100 in savings for a couple. That’s two nights of decent accommodation in a cheaper city, or a lot of standing-at-the-bar lunches.
The deeper point is that this method doesn’t require staying in worse places, eating worse food, or skipping anything. The trade-off is purely temporal - being somewhere on a Thursday instead of a Wednesday. Most travellers have more flexibility than they think, and almost none of them use it this way.
What you can’t fully solve is the crowds. Free entry nights can be busy, particularly in peak summer. Whether that bothers you probably depends on whether you’re the kind of person who’d rather see the painting with fifty other people than pay €18 to see it with twenty.