Both my mares came to me tiny (they were long yearlings). Something around 13'1. QH ended up being 14'3, paint 15'1 (BTW, string measurements gave me exactly those heights
). QH topped by 4 yo (she was on faster side both - mentally and physically), paint by 5 yo. So IMHO right nutrition and enough exercise are the keys to the growth.
In your case since we don't know the history, parents, etc. (and I'd be curious about string measurements too) it's impossible to guess.