Can a person be a master at a martial art at any age?

One never truly achieves mastery of a martial art.
Can you move it like a second language? Yes you can. You can achieve this in ten years through hard work and constant study.
What are you looking at studying?

I haven’t really decided yet, but I am looking into taekwondo, aikido and tai-chi.

Tai-Chi will take the longest out of those three but this is only my opinion. For some, there is something that just clicks, for others it will require more than hard work to truly grip hold of it.
My daughter studies TKD. I’ve studied Aikido and Tai-Chi but not extensively. I grew up studying Greco Roman Wrestling and Southern Wushu with little bits of the Northern style taught here and there. Some other styles I have attempted never really took hold except for Kenpo which was taught in the Army.
There are so many styles of Kung Fu and fighting forms out there. Judo is great, Kenpo is great, Krav Maga is awesome for self defense.
Are you looking for something for yourself or are you looking for something that will allow you to defend yourself?
One of my teachers is married to the Tai-chi Olympic gold medalist. I saw her hit a beam in our building without moving her feet and burst a regular balloon that was taped to the other side of it. It sounded like she was taking the entire building down. Amazing.

Yes, you can start a martial art at age 30 and be able to reach the rank of a master instructor. As @judochop notes, there is always more to learn in any martial art; but under most definitions of mastery, you are not excluded simply because you waited to start. Maybe if you were at an age close to death I’d give a different answer, but I’d still encourage you to give it a try if you are interested.

I am looking to start a new skill as well as self-defense. I am also attracted to sword training,but my only worry is that sword training does not really look like it can be applied practically to modern self defense

@Eggie Let me know how it goes. I’m doing the same thing trying to get into Wing Chun but with kids in the house and a busy family it’s not looking good so far.

@Eggie a lot of martial arts are not as practical as you think in a self-defense situation. It will however teach you how to punch properly and kick properly which can help you greatly in time of need. Strength, agility and stamina above all usually triumph. The best thing to do in the event of a street fight is to run. If you need to stay and fight, I suggest to never stop moving. If you are looking for self defense then Krav Maga is probably your best option and over all it is easy to use. It also applies everyday type objects, like bike locks, blunt objects, bags, rocks and straps with sudden bursts of energy.

Even just a few Krav Maga self defense workshops taught me (and a whole room full of women of various shapes and sizes) many good and very useful self defense moves.

Karate and hapkido can also be good for practical self-defense, but a lot depends on the particular school. If all you were interested in was defending yourself, my suggestion would be to participate in a weekend practical defense seminar and buy a gun. Since you are interested in more than that, however, a martial art seems sensible.

A word of warning about sword training: if you are not training in a martial art that is specifically centered around the sword—e.g., kendo—then be sure to ask in advance how much actual sword work gets done as part of a school’s curriculum. Many of the techniques that became aikido are derived from sword techniques, for instance, but not all aikido schools do much with an actual weapon in hand.

I’ll second @judochop

Tai-chi is the hardest to truly master. However, being more of an internal rather than external art, age is not as much of a limitation as it is in the more external forms.

I once had the opportunity to study a sword form with a group of Wu Dang monks (Yep, the real deal straight from the monastery). During one of the breaks one of the monks was over in the corner with a sword offhandedly fiddling around with it (not trying to do anything strenuous or impressive, just fiddling around). There was a point where his blade moved so fast I literally could not see it (and to him it was effortless)! It was an awesome sight to behold!!!!

I started T’ai chi ch’uan training at 32 and reached Master status in 8 years. So, yes you can.

You obviously don’t know of Morehei Ueshiba, a.k.a. O Sensei, founder of Aikido.

Watch what an 85-year-old man can do. He claimed that as age took away his muscles, he only grew stronger.

Then again, tehr eis more to mastering an art than smooth moves ;)

Watch Bruce Lee and Jackie Chan before you turn 30! Chances are you will start earlier.
Starting at late ages might require you more time to master it. Explanation – : body resistance to injuries reduce with age, that means injuries will take more time to get better.

@jerv But Morihei Ueshiba did not start after age 30. He began his martial arts training in his mid-teens.

@prasad You bring up a good point. One who started earlier will be making fewer mistakes—or at least different kinds of mistakes—by the time he is approaching 30 or above. So how an 15-year-old experiences the training for a particular rank will be different than how a 30-year-old experiences it.

So my advice to @Eggie is to remember that you are training against yourself, not against any of your fellow martial artists. Younger students may race by you. Then again, they might later get stuck on something you breeze through because it is not the kind of thing that can be approached impetuously. In either case: that’s their journey, not yours.

Do you have other sports background? This will make a difference.

My tai-chi master started around thirty, and she’s definitely around master status. I think she does about five hours a day on average though, and she’s now 70. Actually, at least a coupel of sifus I know started tai chi rather late, and after a major illness/injury.

Tai chi takes a while and you will need a sifu who can/will train you in push-hand combat, but in the long run, it’s probably the one that is most nourishing to the body and you can continue it into a ripe old age.