Sure you can. You can make your own travel arrangements, but it will be good if you know the language, as most Thais speak only the Thai language, so moving around Thailand without an interpreter may be a hassle. Good Luck!
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the answer is definitely yes. I don't speak Thai too. But I went to both Bangkok and Chiang Mai on my own. For certain tours, I took an organized tour with a tour guide but for more then 3/4 of the time I was on my own =)