The old adage "The more you make the more you spend" seems to be what happens to most. Everyone lives differently. If you can save a quarter of what you make now, you will probably be able to save when you are on a travel assignment. If you live paycheck to paycheck now, you will most likely continue. It's human nature, history, education, consequences and circumstances. I have been a travel nurse most of my career. Love it. If I knew when I started what I know now I would have made different decisions. Didn't, couldn't and nor can anyone. Read (other peoples experiences), evaluate(how you may react) and plan to do what you really feel you can accomplish. If it will cause you stress, constant worry or you never feel like it's right, then it's time to reevaluate.
Considering the cost of living, the price to travel from one place to another, the multiple licenses, the lack of benefits(maybe) and the sometimes stress of finding an assignment. There is a rather high overhead associated with travel nursing that you won't have if you maintain a full time position with one or two different employers. You get some of that money back in "travel reimbursements" (Same now as it was in 1989 though) and possibly housing stipends (or "free" housing), you just have to remember what someone may say is free, really isn't you are just paying for it in a different way. (Hospital pays company, company pays themselves , their staff, their rent, their expenses, your rent, your expenses and then you get the rest as your hourly wage.) And I think they call that the "Tax Advantage Plan" a way for them to pay the least amount in taxes and you to pay the most.
So in answer to your question 'Can you save money as a travel nurse'? My answer is Only if you can save money with your current job, if not then probably not.