The interplay between artificial intelligence (AI) and psychology, particularly in personality assessment, represents an important emerging area of research. Accurate personality trait estimation is crucial not only for enhancing personalization in human-computer interaction but also for a wide variety of applications ranging from mental health to education. This paper analyzes the capability of a generic chatbot, ChatGPT, to effectively infer personality traits from short texts. We report the results of a comprehensive user study featuring texts written in Czech by a representative population sample of 155 participants. Their self-assessments based on the Big Five Inventory (BFI) questionnaire serve as the ground truth. We compare the personality trait estimations made by ChatGPT against those by human raters and report ChatGPT's competitive performance in inferring personality traits from text. We also uncover a 'positivity bias' in ChatGPT's assessments across all personality dimensions and explore the impact of prompt composition on accuracy. This work contributes to the understanding of AI capabilities in psychological assessment, highlighting both the potential and limitations of using large language models for personality inference. Our research underscores the importance of responsible AI development, considering ethical implications such as privacy, consent, autonomy, and bias in AI applications.