What percentage of systematic data starts to be considered random
Tests of randomness like (my favorite) Kolmogorov complexity will tell you how random those objects seem scientifically. (I'm not certain but I think your describing chaotic behavior rather than random behavior because the random objects can only be rearranged rather than replaced.)

Aside from mathematics this is an active area of psychological research. Those with disillusion illnesses like schizo typical often have the trouble identifying when a pattern is random. The sick brain seeking to bring the universe into order or create a new reality to match its broken perceptions deludes itself into believing that random events happen as per a control. This is called Pareidolia (when there is something which might vaguely resemble a pattern) and Apophenia (when the data is obviously random). 

Your test would be measuring Pareidolia, Apophenia as well as Pattern recognition.