why might some people refuse speech and language therapy?

They can refuse if they don't agree with the results of testing by the SLP. Just as parents may not want to stigmatize their children, adults with strokes/neurological issues may have little or no insight into difficulties that they may have. Another issue is lack of understanding of an SLP's role not just speech but also cognition and swallowing, so they may say "I talk fine" when that was never the issue.

People have the right to make informed decision about their care, just as you or I could refuse the flu shot or don't want to take a multivitamin.

My son is 2.5 and until a month or so ago said VERY little. I could tell he was just taking his time. He babbled constantly, but chose to focus on learning about other things rather than talking. When he did decide to speak, we found that he knew tons of words, including dinosaur names. He will repeat anything you ask of him and follows direction very well. He can also count to ten and say most of his colors. All of this came out of the blue one day.

We had someone review him for speech therapy back when he turned two, but decided we didn't want to stress him with it. His hearing was fine, and he could say all his vowels and consonants (we would hear them in his babbles) so we didn't stress over it and let him do it in his own time. He still isn't saying the 'normal' amount of words for his age, but he is definitely saying words that most his age don't. So we choose not to give speech therapy because we just didn't see a need to stress over it. In our case, our son was just taking his own time. Why force him to develop on our time line?

Many parents refuse any and all testing and special services for their children, afraid they will carry a very negative label for life and be considered unintelligent.

Some people are comfortable with how they are since they know how to act and think like that, and they may feel worried about the idea of change since it is unknown to them, and they also do not know how other will react to them.