I am asked this question a lot. Most times, I find that the person asking it doesn't really want to let her husband go (deep down) and wants to know something else. What most people are asking is: "Can one person save a relationship?" or "Can I change my husband's mind and save the marriage?" In essence, the person asking a question like this wants to know that if in the future, if things can be turned around or will there be a divorce. I know the answer to these questions is a resounding yes because I've lived this situation and have come out stronger on the other side. I am still married today because I refused to let my own husband go. (More on that here.)
Every situation is different of course. The answer to this question will depend upon a lot of things, such as the reason the husband wants out, how long the relationship has had problems, and how willing both parties are to resolve it. However, that being said, it's entirely possible to save a marriage when you are the only one who wants to, but you'll probably need to take calculated baby steps, with each success building upon the next, to do so. And, eventually, you'll need to address the underlying problems that got you here (although I don't recommend this until your relationship is back on firm ground. If you try to have serious discussions before this, it could drive your husband further away.)
Why Husbands "Want Out": What can sometimes make this problem more complicated is that husbands will typically give you vague answers about why they are unhappy that don't tell you much. While sometimes there is a clear reason for wanting to split up, (like a major disagreement, infidelity, or a huge family crisis) often the reasons are vague and the husband will say something like he's "fallen out of love," he's "just not happy," etc. More, his body language and / or actions may be completely contradictory. He may hold you, tell you it's ok, act affectionate one minute and shut you out the next. So, it can be hard for a wife to "read" this situation and determine her chances of saving the marriage.
Understand that, barring the clear reasons mentioned above, husbands typically want out because they've lost feelings along the way, but also understand that, the feelings are typically more about themselves than about you.
When you were first dating and falling in love, the quality and quantity of attention that you both paid to one another likely propelled very positive feelings in your husband. He likely felt attractive, interesting, worthwhile, worthy, and intelligent because you thought all of these things about him.
Of course, somewhere along the way, we all get more and more responsibilities. We have to care for children, devote more attention to our jobs and caring for our homes (and possibly our aging parents). There's only so much of us to go around. The time we have to devote to our relationship suffers and as a result, the cracks begin to show. Most of us make the mistake of assuming that our husbands know we love them and would give them more of our time and attention if we could.
The problem is that, although husbands know our intentions are good deep down, they still wish the situation was much different. They still need us to validate them, and just like us, deep down they want to be loved and shown appreciation regularly. When they feel this isn't happening (even if the intentions are there), they're typically bitterly disappointed because all of the good feelings that this cycle generated about themselves in the past are now gone. A husband's "wanting out" is typically his concession that the feelings you used to be able to generate in him about himself (because of the intimacy between you) are now gone, and he isn't sure about how to get them back or doesn't believe they'll come back. (Your job is to show him they can. But you'll often need to do this very gradually. I learned this the hard way. There's more here.)
How To Get The Original Feelings Back And Save The Marriage: Of course, the obvious thing to do is to return the original feelings that made your husband happy in the relationship. Now, having said that, I have to stress that you can't be overly obvious about this. If your husband gets wind that your efforts are fake or he thinks you're game playing, he's likely to resist your efforts even more.
What is important is that every opportunity you get, you present yourself as the kind, intriguing, happy go lucky, interested, and loving person he first fell in love with. I know it's very hard to do when your heart might be broken, and I know your inclination may be to repeatedly question him and attempt to get relief from this situation as soon as you possibly can by bombarding him with communication, but all of these are mistakes because both are born out of desperation and will make you appear unattractive and needy. (I'm not judging. I did this too.)
You want to portray yourself as the exact opposite of this. It takes discipline and it takes work, but in the long run, it's the only way I know to get the original feelings back in such a way that will feel genuine to your husband and will not elicit more of the bad feelings he may have right now.
When Is It Too Late To Save The Marriage?: It's very rarely too late. If you still have a desire to save the marriage and your husband isn't being (or wasn't) abusive, then there is always a chance to save the marriage. However, if the relationship is very damaged, more baby steps (and more time) are generally required. I have a lot of people who say, "Yes, but my husband says he hates me. He's really mad." I usually answer that is good news. Because hate and anger are strong emotions that are not so far from love. If your husband didn't love you, he wouldn't have these strong emotional reactions. He'd be indifferent instead. Indifference (on the part of both spouses), more than hate or anger, is usually more indicative that the marriage is really over.
My husband said he hated me a couple of times. Thank goodness I didn't let that stop me. I kept right on working on the marriage (by myself, since he wasn't interested at first.) Eventually, (though commitment and lots of effort), I was able to not only save the marriage, but make it stronger. So, it was very much worth the effort. You can read my of how I stopped the divorce on my blog at http://isavedmymarriage.com/
There are links to more articles about saving your marriage at http://isavedmymarriage.com/