Yes, but you can speed up the process a lot by learning from others. There are many great harmonica tutorials on youtube, for example. Different teachers work on different styles of playing and different genres of music, but the most common is blues harmonica.

A diatonic harmonica's simplicity makes it a great choice for the beginner. Most harmonica teachers recommend starting out with a 10-hole diatonic harmonica tuned to the key of C. The Hohner 1896 Marine Band 10-hole diatonic harmonica in the key of C makes a great beginner's instrument.

With regular deliberate practice, you can expect to be playing simple pop tunes within around 3 months. Within 6 to 12 months, your technique will improve and you will probably be able to work on bending notes (a very important skill for getting the best out of a harmonica).