Is it reasonable to apply for (postdoc) positions early? When preparing an application for a position, it goes without saying that everyone wants to make the strongest possible case for themselves. Hence, it seems that the best strategy would be to wait until the last possible moment - just in case in the meantime an acceptance letter comes from that journal where you submitted the paper half a year ago or your latest project reaches a stage when it can be posted on arXiv.

However, I am wondering if this is reasonable. Presumably, whoever is assessing the applications has internet access and can easily check if there are any updates (say, by visiting the personal webpage or googling the applicant's name). Since application procedures tend to be somewhat lengthy, I would also imagine that hiring committees are well aware that the applicants accomplishments can increase in the meantime. 

[The position I have in mind is maths postdoc, but I welcome a broader perspective.]
My recommendation is first that you don't worry too much about it, but I think there is an important exception: if you are applying using MathJobs, then you should apply once you have your documents ready, hopefully at least a couple of days before the deadline.  In particular, you should not wait for your references to add their letters if that is the only thing you're waiting for.  Those will show up when they are available, but it's much better for someone to see your application is there without all the letters, than for you to get lost right in the avalanche when everybody else submits right at the deadline.  Similarly, you can update your CV, so there's no reason to wait in hope of another paper to getting accepted.