I spent a fair amount of time earlier today trying to set up todo.txt with Cygwin on my Windows 7 boot. For the longest time, I was getting "Permission Denied" whenever I tried to write to files inside of Dropbox.
This was a result of some misconfiguration of cygwin and Dropbox; long story short, if you run into this problem, chmod-ing the files with the necessary permissions didn't (sadly) work. Stack Overflow suggests playing around with /etc/passwd inside of cygwin, and even offers a pretty cool guide about how Cygwin maps Windows to POSIX-like permissions.
Ultimately, the solution that worked for me was the simplest one: in Windows, I opened the file and gave everybody full permission for it.