Can they be different from each other and still work together?
You bet. And if you're not fortunate enough to find a magnificent vintage danish modern table (or fill in the blank with whatever your dream vintage table is) with 4 to 6 matching chairs and a leaf in mint condition at an affordable price, well, then you have to get a little bit creative.
Rule Number 1. Buy something in the same shape and size as your dream table, in moderate vintage condition. And buy just the table. Don't worry about the matching chairs. Chances are you will get a better "deal" on the table without the chairs. I notice that we all tend to shy away from tables without chairs. I believe this is due to the fact that we know it won't be easy to find the chairs and we love easy. Easy is a thing for us these days. i.e. the Staple's Easy Button.
Rule Number 2. Look for chairs that have the general shape and feel that you are looking for. Not ones that match the table perfectly and here is an instance where I would suggest ignoring the details. Always remember that chairs are far and away the easiest thing to alter. They can be painted with spray paint and often times changing out the upholstery just means turning the chair over, unscrewing the seat from the chair and putting on new fabric. Easy peasy.
Rule Number 3. There are no rules. Do what you want. Have fun with it. It's a place to eat and enjoy a little chit chat and time with your family and friends. Relax people.
I gathered a few of my favorite images for mismatched dining room chairs and tables.
This works so well because the home owner choose 4 chairs from the same era as the dining room table. Then she painted them and recovered them in a modern fabric. I'm in love with this whole house, inside and out. Check it out here.
This is Elsie's dining room from her blog called A Beautiful Mess. She is quickly becoming one of my most favorite bloggers. This is so creative. Such a variety of chairs and she makes it all work by painting them the same color. Love it.
Another creative solution is to use stools, ottomans and/or benches as seats for your dining room. Image from House Doctor, seen on Decor8 here.