I'm pretty sure their chances are just fine. :) They are still making hires which I'm pretty sure they wouldn't do if they saw their company declining. Also trello and asana attract very different markets of customers I'm sure compared to 37signals core demographic. Take a look at the featured customer videos at basecamphq.com. It's wedding planners, brand design, a self help firm. Not software developers who seem to be the folks gravitating to trello and asana.

If my business is threatened by several competitors one of the things I might do is hire new people to work on new initiatives.

I didn't find Trello very useful or easy to use. It's a new way to do things but probably not the ideal way to do them.

True. At this very moment using Trello has made one ongoing project very hard to manage.

Trello and Asana are free (last time I checked). It'll be hard to make money from people using their apps for free. No money = no app improvement. IMHO.

perhaps, they will start charging for larger teams, or advanced features.

Both companies are well funded (and founded by wealthy people), so I do not worry about the 'no money == no app improvement' case

Don't forget about http://apollohq.com

Why would I get downvoted for sharing a product I use? I am in no way affiliated with the team or product.