sixofhearts
This seems to be independent from your question. Everyone on HN can come up with a reasonable speculative answer to your question (and mine would be a "yes"), but focusing on the issues makes clear if it's even worth competing in the first place.
With the information you provided, you seem to have a solid target market identified. How many teams are there that, like yours, that could benefit from a Basecamp alternative? And which of your team's needs aren't met with Basecamp, and is it something that Basecamp could replicate quickly? Dig a little bit deeper and discover the specific needs - that should reveal more useful information than any of us could provide :)
Ask HN: What's the first thing you do when you wake up?
I've found that the way my day turns out is largely dependent on how I start it.My start usually involves: getting out of bed, meditating, making coffee, using the bathroom, and some reflecting (what I'm thankful for, what I like, what I'm doing for the day). When I start like this, my day tends to be fulfilling and productive. When I don't, it's a crap shoot, but sometimes it still goes...
-Sam's Teach Yourself Unix in 24 Hours
-Sam's Teach Yourself Shell Programming in 24 Hours
These won't make you a shell genius immediately, but they'll create a good basis for future, specialized learning.
Also, I try not to be a material snob, but buying the book instead of using a website or .pdf file made a world of a difference (especially during a morning commute).