This paper is devoted to the study of cognitive and personality characteristics of intellectually gifted students with a different ratio of verbal and non-verbal abilities at various secondary school grades. The sample included 280 students of V (aged 11-12), VI (aged 12-13); VII-VIII (aged 13-15) and IX-X (aged 15-17) grades of Moscow gymnasium. IQ of all students was above 115.Three approximately equal groups of the students were selected in each age: with relative lag of verbal abilities; with relative lag of non-verbal abilities, and with equality of verbal and non-verbal abilities. The Cognitive Ability Tests (CAT), Raven Advanced Progressive Matrixes (APM), the verbal and figural Tests of Creative Thinking (TCT), the questionnaires on anxiety and academic self-concept, and school marks were used. The results obtained demonstrated the certain school difficulties in the students with high general intelligence, but relatively lower non-verbal and especially verbal abilities. Distinctions among the students with the different proportions of verbal and non-verbal abilities observed not only on the CAT-variables, but also on the APM and TCT, academic achievement, as well as personality characteristics. It was shown, that the combination of advanced intelligence and school underachievement could result in the students with lower verbal abilities to inadequate academic self-concept and increased anxiety, and then to personality and disciplinary problems as well. So, self-estimations of abilities to some school subjects, especially to Russian and Foreign languages, in this group were significantly higher, despite the poorer school marks, than in their classmates. The obtained data allow to assume, that a contradiction between high academic self-concept and lower achievement can be a source of the increased anxiety in the gifted students with relative lag of verbal intellectual abilities. This study was supported by Russian Foundation for Humanities, project № 14-06-00564.