I work for a big for-profit online university, and we see this problem all the time. It's a big deal for us because we're for-profit, and when a person can't attend our school because their HTTPS connections keep dropping, it cuts into our profits.
I'm going to be upfront though; I am not a networking person. I do not have an understanding of TCP/IP protocol.
That being said, I'm pretty sure this is what is happening:
Most wireless connections are encrypted. That is, encrypted while your data is traveling through a radio signal between your computer, and the access point or router, where it is de-crypted, and passed along completely unencrypted through the rest of its journey.
That last part is normal. Wired network connections are NOT encrypted. This is not a problem for the most part, because nodes (the big servers your data hops across on its way to its destination) have SO MUCH traffic moving through them, an unscrupulous employee with a high level of access would have to fish out your data from the stream, and your data is probably no more interesting than the next guy's data. (Sorry.)
However, it is because of that remote possibility, that we use encrypted secure connections for bank transactions, VPNs, email, and connections to our colleges and universities websites. That way the unscrupulous employee would have to fish out your completely nonsensical data, and spend the next 2,000 years using a supercomputer to figure it out.
So here's where the dropped or timed out connection comes into play. This is kind of difficult to explain using plain language terms, so techies please bear with me:
First, you have to understand that encryption increases the amount of data you are transferring. Say that you were going to encrypt my name, Seamus. The encrypted version would not be six characters long. The encrypted version could be 60 characters long.
Ever notice how sometimes on a secure site your web browser will say "this page contains both secure and non-secure items. Do you want to display the non-secure items?" You get that message because the web page you're on does not want to encrypt the transfer of images between you and their server. One little image can easily be equal to the data of all the text on the page. Encrypting the images would make the page load on your computer EXTREMELY slowly.
When you are on a secure (aka encrypted) wireless connection, and you try to access a secure website, you are overlapping encryption. At that point, you are sending a massive amount of data between your computer and the wireless access point. Just a few dropped packets or network interference (even if you have maximum signal strength) can bring your connection to a crawl, or time it out completely.