SourceTree limits how much of a diff it displays. If there are too many changes, SourceTree will eventually truncate the display in the middle of a hunk. It does not let you stage or discard that hunk because it is not showing you the entire hunk.
If you choose to stage or discard earlier hunks, it should reduce your diff content enough to display that entire hunk, which would let you use the buttons.
It's an annoying feature, but I assume it is necessary to keep SourceTree responsive when viewing large changesets.
EDIT: If your hunk is still too large, you can try highlighting large portions of it by clicking and shift-clicking within the large hunk, then staging or discarding "selected lines". I have never tried this in a partially displayed hunk, so I'm not sure if you'll be allowed or not.
BTW I am using Windows 7, SourceTree 220.127.116.11 and git version 1.9.0.msysgit.0 Seth, I tried your suggestion but to no avail. The last hunk of about 500 lines, a pure insertion, still had the stage hunk button greyed out. 500 lines does not seem unduly large for a hunk. David
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