Trello is designed for a high level overview of a project. So if you start to get too many items on your board, either it's not the right tool for what you are trying to do, or you need to think more about what you are adding to Trello. For example, you could use Excel to take notes, or make a todo list, but it's not the right tool for that. Trello really shines when you are using it for group collaboration and either focusing on a high level (and leaving the details to something else) or only focusing on the very top level important details.
I know a lot of companies that use Trello in tandem with tracking apps. There's even a nice bookmarklet that integrates with a bunch of them (FogBugz, Jira, Github, Saleforce) so you can hit one button and turn your case into a card in Trello. See https://github.com/danlec/Trello-Bookmarklet But don't make the mistake of trying to duplicate what those programs do in Trello.
We meet regularly with other departments to triage their request queue (list) and move it forward. It's great for them to see the software process and see their request in context with the (many) other requests.
We often break out separate project boards and link to it from a master (planning) board. That really helps break down the larger tasks.
mhp noted that it's not designed for bugs, but we have no problem using it for that, in conjunction with some GitHub issue tracking.
I'm certain a startup would have no issues adopting it into a great workflow.
I think Trello looks nice(r), but it's easy to create a mess with it.