The 4226 event is not related to boot times only (or, at all). It means the number of connections attempts in a certain small period of time (as in "per second") are "too high" for the limit already set in Windows.
MS originally limited this number in Windows to reduce the chances that some malware would spread (communicate) using your system (this is a simplified explanation, but it's enough in this context).
You could change the limit, so the 4226 event won't be displayed in event viewer, but the number of connections will still be the same. Being more an information than a real problem, I wouldn't care much about it, specially if this only happens once in a while ("low" frequency).
If this event really happens only at boot times, then "too many" tools are trying to connect at the same time when you boot. You could check which auto start up items you have, which ones are really necessary, which ones are trying to connect...
Particularly, in avast, you can configure it to reduce the number of connections, according to the type of network connection you have. Try setting avast to connect with "dial up only" (even if it is not really dial up). Other settings in avast may play a factor too. And you can also manually configure avast to delay the first connection (by editing the ini file).
If the event happens in additional situations (not only at boot time), then you would need to identify which specific software is trying to connect "too much". Typically, P2P applications are the most common ones generating this 4226 event. By changing it's settings you reduce this 4226 event (and you may gain in P2P speed too).