I've been doing something similar for the last couple of years (SVN and Ubuntu One rather than Git and Dropbox). In general it works very well but you need to ensure that both computers have properly synced before you switch working between them or you risk issues with conflicts and/or corrupted files. Also you should have a backup plan in case things go wrong. In my setup I only keep the working copy in Ubuntu One with the repo somewhere safer. This means if something doesn't work properly I can simply create a new working copy. I realise Git works a little differently but it is always worth ensuring you can recover from failure as Dropbox and equivalents are all slightly less reliable than traditional file storage. The final thing to bear in mind is that if this is for work, some employers will be unhappy about their proprietary code being hosted on the "cloud" rather than on computers they control.
I was having a dark day and after a weekend off have realised my startup is not so doomed after all. However, many thanks for the tips - hopefully they'll be useful to others in a similar situation.