Question: I recently have had a discussion about my friend's tone in his email.

His supervisor asked him whether he wants to attend a summer school in May. He wanted to thank him for his invitation and also wanted to keep the mail short. As a result, that is what he came up with:

> 
> Hello Professor,
> 
> 
> I have plans during that period. Thanks, but no thanks.
> 
> 
> Regards,  
> 
>  John 
> 
> 
> 

He is getting along well with his supervisor, but I believe that the phrasing is a bit disrespectful nevertheless. It might be the case that I am not a native speaker. 

Am I exaggerating the situation and this is something can be overlooked?

Answer: 
'Thanks, but no thanks.' is sarcastic and rude. Your friend should NOT use such wording, ever. If he has, he should personally visit the professor and apologize for having done so. If he is not a native English speaker, he might be able to blame it on that. If he is, he should have known better, and will have trouble explaining his use of such a phrase.
If I was invited to attend something that I couldn't I would simply say, thanks for inviting me, but I'm sorry I'll have to decline, as I'm busy that day.