Funding and pressure to finish quickly. I have pretty good resources from my PhD program, but there's constant pressure to finish quickly instead of taking the time to go deep and really explore the questions wherever they take me. And my ability to follow questions wherever they take me is greatly limited by how many grants i can string together to fund the necessary travel (and grant applications take a lot of time to write, leaving even less time to dig into my sources). It's a challenge almost everyone is facing right now, as Humanities budgets are being cut across the country.

This article sums of the obstacles I experience re finishing quickly pretty well: http://www.thismess.net/2014/11/finishing-humanities-dissertation-in.html

I also have difficulties with limited textual sources (I study early medieval history, not much writing survives), but I consider that more of a challenge than an obstacle, as it merely forces me to find creative solutions rather than preventing me entirely from doing my work.