False - it's the other way around. An organism has two alleles for each trait, for example the gene for blue eyes is given by two recessive alleles, bb, while someone with brown eyes might have the alleles Bb.
Now remember that two gametes, or sex cells, form an organism, so they each have to have one allele per trait in order to create an organism. So if two gametes, one with the eye color allele B and the other with allele b fertilize each other, they will produce an individual with Bb alleles.