Nowadays, when the migration processes have contributed to the high diversification of societies, one can observe that many social environments put extra emphasis on actions aimed at the integration of minority and majority groups. One of the issues connected with the integration of socio–cultural migrants with the host societies is their acquisition of intercultural competence. Having appropriate intercultural competencies increases the probability of success in the integration processes. However, this does not mean that the immigrants’ opportunities for integration in the cultural and intergenerational dimension are not problematic. Those difficulties are related to the multi-faceted cultural diversity of migrant groups recognizing different norms and values, what in a consequence may lead to tensions and even conflicts. It should be emphasized that intercultural competence and its implications – the success of integration or the integration failure – do not depend only on cultural diversity. What is important is the age of the migrants in addition to other psychological factors such as awareness of one’s self-centredness, attribution error or the group anxiety. The paper focuses on the acquisition of intercultural competences by the migrants in the integration process with the majority group depending on the migrants’ age and cultural capital. From the theoretical point of view, the younger the migrant is and the more similar to the new social reality the cultural context of the migrant’s is, the higher chance is to acquire the necessary cultural competences. It is indeed related to the new reality of the migrant’s life. Moreover, the level of one’s awareness of one’s self-centredness and the group anxiety as well as one’s perception of the reasons for one’s own and other’s behaviours (attribution error) can diversify the process.