Graduate training is a key element in producing a scientific workforce that reflects the nation's diversity. This paper examines data from a 2013 American Chemical Society (ACS) survey of 2,544 chemistry masters and doctoral students and reveals barriers to reaching this goal. Multivariate statistical analyses indicate that women reported significantly less supportive relationships with advisors. Women were less likely to plan to finish their degrees, and for PhD students, the discrepancy was larger for students at the start of their graduate program. Women were also less likely to pursue the next level of training, and the gender difference related to postdoctoral plans was greater for those who identified with a racial-ethnic group traditionally underrepresented in chemistry (underrepresented minority, URM). URM students who were beyond the first year of their graduate program reported significantly less supportive relationships with peers. They were also less likely to have funding sufficient to meet their needs and more often used personal resources including loans. Despite these difficulties, URM students were more likely to definitely plan to finish their degrees, and men who identified as URM were more likely to plan to pursue postdoctoral work. Independent of gender and identification as URMs, students in more highly ranked schools reported less advisor support. Extensive open-ended comments indicated that large proportions of the students desired more attention and meaningful feedback from advisors and changes within their programs to promote support for students and advisor accountability. Suggestions for future research are given, and a companion commentary discusses needed directions for change.