I'm flying on United with a bunch of tight connections. I also have travel insurance which covers missed connections because of delays due to weather or mechanical failures, but they require proof of this in order to submit a claim. I know airlines have a tendency to blame everything on weather, but is it easy to obtain written documentation to this effect?

Also, I have some very tight connections that the slightest of delays could cause me to miss. At what point (how many minutes) does a delay constitute a delay?