Optimal language acquisition via reading requires the learners to read slightly above their current language skill level. Identifying material at the right level is the essential role of automatic readability measurement. Short message platforms such as Twitter offer the opportunity for language practice while reading about current topics and engaging in conversation in small doses, and can be filtered according to linguistic criteria to suit the learner. In this research, we explore how readable tweets are for English language learners and which factors contribute to their readability. With participants from six language groups, we collected 14,659 data points, each representing a tweet from a pool of 4100 tweets, and a judgement of perceived readability. Traditional readability measures and features failed on the data-set, but demographic data showed that judgements were largely genuine and reflected reported language skill, which is consistent with other recent studies. We report on the properties of the data set and implications for future research.