It's certainly possible, but depends on many things. I try to travel as cheaply as possible while still traveling exactly how I want to. This can mean traveling on a budget to Grand Floridian.
1. Tickets are pretty much a fixed cost. You can buy them through Disney or a discount site like
Undercover Tourist for a slight discount. If tickets are at a deep discount, chances are it's a scam. You should decide what type of tickets you'll need.
2. Decide if you are staying on property, semi-on property, or off property. With one exception, I've always stayed on property. I love the convenience and the Disney "bubble". Once I land, I don't want to drive, so included magical express makes it feel like my vacation starts immediately. There are semi-on site hotels (Swan, Dolphin, Wyndham) that will give you proximity for a lower price, but I've always stuck with Disney.
3. If staying on site (or buying tickets through Disney) use Disney gift cards. You'll get a discount at certain bulk stores (BJs.com has $100 for $95 and you can get free club trials). You can also use credit card rewards - I always use my 2% cash back card but discover will sometimes have 5%. Also, I load the gift cards into the Disney Vacation Account for an extra 2% in Disney gift cards back.
4. Disney's prices aren't always the cheapest for packages. Orbitz often has coupon codes that stack on the Disney discount, but you can't pay with Disney gift cards. One other
travel agent I've used has "magical deals" that can give you a 10-20% higher discount on hotels than Disney, but I won't go into specifics here. You might need to google that.
5. Money can be saved while at Disney too, without sacrifice.
Amazon Prime Now delivers to Disney. Don't pay Disney prices for bottles of water! If you aren't a big eater, or aren't too hungry for a meal, you can always order QS meals off the kids menu. Often it can be more than enough and you'll save a few bucks. I love the Dining Plan, but it usually doesn't save money. When I don't get it, I load up a Disney gift card ahead of the trip to use on all meals. You can find menus online and look at items you'd probably get to get a good idea of how much you'll need. Gift cards can be used for tip and disney restaurants too. Get the gift cards discounted ahead of time for more savings!
6. Time of the year matters! Hotel rates and airfare rates vary greatly depending on when you travel. If kids are out of school, it's going to be more expensive. Early February and early December are good examples of cheaper times to visit. Plus you'll hopefully have less crowds to deal with.
Good luck in your planning.
Edit: To give a concrete example, my wife, baby girl and myself will be going at the end of August. The baby is free. We are staying 6 nights at the beach club for ~1550. No tickets (would be ~$900 for hoppers) since we have APs. We'll fly on points, but that would be ~ $500. I also got gift cards through target for the room plus our APs during their 10% off sale, so ~15% should be subtracted for the discount, cash back, and DVA rewards.
So at a glance total for the 2 of us plus baby would be cash ~$2950. $2450 of that could be paid with gift cards at a discount of at least 5% if you use BJs, plus cash back. This doesn't include food, but is at a Deluxe resort. You could very easily stay at a moderate (I recommend POFQ) or value (I recommend AoA or Pop) for a much lower hotel price.
It takes a bit of work and research to save the most money, but for me, that's part of the vacation fun!