This is just a single example, but "height" has received a lot of GWAS (genome-wide association study) attention, and last year the GIANT consortium published results of an analysis on 250,000 people looking at the genetic influences of height .
Using just 2,000 genetic variants they could statistically account for 20% of the phenotypic variation in height. Using 9,500 genetic variants this rose to 29%.
What this emphasises is that common traits (such as height) are highly polygenic; there is no "height gene" that controls it - instead it is the combination of many small-effect variants that cumulatively give rise to an individuals final phenotype (height, in this case).
Pigmentation (of eyes, skin and hair) on the other hand is not nearly as polygenic. A 2013 review  discussed the known genetic variation for these traits. Prediction of eye colour using genetics, for instance, requires only six genetic variants "to be suitable to predict blue, intermediate, and brown colors with an overall accuracy of 0.91, 0.72, and 0.93, respectively".
So in contrast to height where even with 9,500 variants measured you would still only have 29% accuracy, you can be 91% sure that someone will have blue eyes if you know the individuals genotype at only 6 positions.
So in short the quote you reference is incorrect - we can predict many common traits with reasonable accuracy - although for some phenotypes the individual predictive capability of a specific genotype may be around 0.01% (e.g. some of the height variants).
- Wood, A. R. et al. (2014) Defining the role of common variation in the genomic and biological architecture of adult human height. Nat Genet 46, 1173-1186, doi:10.1038/ng.3097
- Liu, F., Wen, B. & Kayser, M. (2013) Colorful DNA polymorphisms in humans. Seminars in Cell & Developmental Biology 24, 562-575