Thank you for all your answers, highlighting very different aspects. They really got me thinking.
I guess it's more about personality and strengths and weaknesses (both my own and my pupils').
I don't think that I'm generally bad with 6-8 year old beginners, otherwise I wouldn't have that many, including one that announced that she wants to be a violin teacher when she is older. With my younger home pupils, I have most parents sitting in the lessons and I guess they'd withdraw their children if they weren't happy and found that I'd put their children off. Yet I'm still teaching many of those former 6-8 year old beginners years later.
Also, I think I can tolerate practice with not much talent and some talent with not much practice, in any age group. If there is neither of the two combined with a complete lack of interest in the instrument, no parental encouragement whatsoever to practise at least a little bit occasionally, yet the same parents push for the next grade, it becomes really difficult.I am encouraging and patient and appreciate that my pupils are as different as it gets, but I'm not a cheerleader or a magician..
If a parent informs me that a child has a particular learning disability or similar, again, it's different as I then know the reason why a child might struggle with something and try to adjust my approach and expectations better than when I'm left in the dark about it and can only guess.
Practice: I'm very realistic about how much energy it takes to get children to practise, how little time there is sometimes, and sometimes the children don't want to stop lessons because they do like the actual weekly lesson, but don't want to practice at home. But a minimum amount should be possible.
lindaff: You are right, the one pupil I had in mind had lessons for a year with another teacher but had forgotten almost everything. The child couldn't play even the easiest tunes in the book, even though (strangely) much harder ones had already been ticked off by the previous teacher. I realised (just counted) that the child had only 6 lessons with me and maybe she still has to get used to me. This pupil has already improved her bow hold a lot which is quite an achievement and I realise I have to give her even easier pieces and slow down a lot. I think because of the pieces ticked off exactly a year ago by her previous teacher I simply expected way too much.
Left-handed violin pupils: Ironically, two of my pupils you can described almost as "prodigies" are left handed (they started aged 5 and 7 with me respectively). They may be the exception that proves the rule though, I don't know? My "problem" pupils are right handed, maybe that's the problem?
Singing and successful violin playing (i.e. in tune): The more experience I have, the more I'm struggling to see how the two should be related. In theory, yes, the violinist should imagine everything at least with the inner ear (maybe here is the clue, inner ear, not necessarily sing aloud?).
But I have pupils who can play beautifully in tune but cannot sing in tune, and can't sing back a note played on the violin or the piano. They sing sometimes a whole tone higher or lower but not consistently so, often they sing something else altogether! But they strangely play in tune or at least correct themselves when playing a note out of tune. Not when singing though.
Yet others can sing beautifully in tune, but play so completely out of tune it is unbelievable. I had a pupil who could sing her violin pieces beautifully in tune, but although she could physically play the violin (and yes, even keep the bow on the correct string) it was often completely out of tune, and she didn't notice or correct even though she knew how to make a note sharper or flatter on the violin.
Quite baffling. If anyone has any ideas why that is and what to do, I'd love to know. I would definitely not choose a violin pupil based on their ability or inability to sing because of the above. Probably making them try the instrument is the only way.