The study is aimed at exploring Master's students' sense-making of various second language (L2) texts and its correlation with interpretation depth and the students' intercultural competence. The significance of this research lies in the fact that it provides an insight on culture-specific issues challenging for advanced and proficient Ukrainian learners of English when they make sense of L2 vocabulary, interactions, public, mass and social media discourse. The study is based on a questionnaire with open-ended questions filled in by forty Ukrainian Master's students. The subjects were asked to interpret several English vocabulary items, idioms, linguistic landscape signs as well as mass media and social media texts. The findings suggest that the students' intercultural competence rests on the information obtained from English textbooks as well as from current mass and social media content. When facing ambiguity or uncertainty, the students tend to show a lack of cultural awareness and resort to first language (L1) assumptions in their sense-making, which may entail a wrong or inaccurate understanding. The students' interpretation of public, mass and social media texts reveals their insufficient background knowledge of L2 culture, which does not lead to inaccuracies in sense-making but prevents comprehensive understanding of messages. The study argues that Master's students majoring in English need to enhance their cultural awareness, to develop their intercultural competence and to be prepared for continuous learning of L2 cultural schemata.