Work on instruction-tuned Large Language Models (LLMs) has used automatic methods based on text overlap and LLM judgments as cost-effective alternatives to human evaluation. In this paper, we study the reliability of such methods across a broad range of tasks and in a cross-lingual setting. In contrast to previous findings, we observe considerable variability in correlations between automatic methods and human evaluators when scores are differentiated by task type. Specifically, the widely-used ROUGE-L metric strongly correlates with human judgments for short-answer English tasks but is unreliable in free-form generation tasks and cross-lingual transfer. The effectiveness of GPT-4 as an evaluator depends on including reference answers when prompting for assessments, which can lead to overly strict evaluations in free-form generation tasks. In summary, we find that, while automatic evaluation methods can approximate human judgements under specific conditions, their reliability is highly context-dependent. Our findings enhance the understanding of how automatic methods should be applied and interpreted when developing and evaluating instruction-tuned LLMs.