OMG, this was one of the most insanely frustrating issues I've ever had with my iPad, but (fingers crossed) I believe I might have resolved the issue -- at least in my case, and hopefully in yours. I use the iPad Air email app to view multiple Yahoo, Gmail and Outlook accounts, and never had any problems until yesterday. Suddenly, in one one of my Yahoo accounts, I clicked the Inbox and received the dreaded "Cannot connect to server" error message. I could still see all the folders in the email account, but every folder said "No Mail" in the left-hand window and "No Message Selected" in the right-hand window. To confuse matters further, I discovered that I could send emails from that Yahoo account, but could not receive them; also, the Reminders from that account were showing up in the iPad's Reminders app, so there was presumably some communication occurring between the iPad and the Yahoo account.
I spent 5+ hours on the phone and online with AT&T/Yahoo technical support last night. They were polite, but utterly useless. They simply muttered the same scripted responses over and over about using their POP server instead of IMAP, and ultimately told me there was nothing more they could do because they aren't experts in "specialized" equipment and email clients. (Perhaps someone should inform them that iPads are pretty mainstream these days.)
Today, I called the Apple iPad technical support number and spoke with Kevin. What a better attitude than the three AT&T/Yahoo drones I dealt with last night! Although Kevin couldn't solve the problem, he listened carefully, understood the extensive troubleshooting I had already done (which were all the steps he said he would have tried), and ultimately set up a call for me to speak with a "super-escalation" technical support person tomorrow to back up and restore my iPad (which is a somewhat harrowing and time-consuming ordeal under the best of circumstances).
Since I figured I wasn't going to be able to concentrate on much else, I continued brainstorming and I seem to have resolved the problem. (Nevertheless, major points for Kevin and Apple for not just giving up on me, as AT&T/Yahoo had done.) If this solution remains stable today, I will give the "super-escalation" person the details tomorrow so Apple will be aware. Hopefully, that can save others the hours of frustration I faced resolving this.
Usual fix: The usual fix for this problem is to remove the "offending" Yahoo account from the iPad (Settings --> Mail, Contacts, Calendars) and then re-install the account on the iPad (Settings --> Mail, Contacts, Calendars --> Add Account). This did not work for me. As before, the folders appeared (after about 5 minutes) and the Reminders worked, but I couldn't see any emails and kept getting the "Cannot connect to server" error message when I tried to refresh the Inbox or any folder.
Second idea: When re-installing the Yahoo account (Settings --> Mail, Contacts, Calendars --> Add Account), instead of selecting "Yahoo!" as the account type, I selected "Other." (This does not provide all the functionality of adding it as a Yahoo account, but I figured it was worth a try.) On the next screen, I entered the Yahoo account credentials, clicked "Next" (at which point the iPad verified the account info) and then clicked "Save." This "Other" account then showed up in the iPad mail app. I could send and receive emails and see all the folders. I then noticed that some of the folders would download and display their contents, while others would not (they just stayed empty). That seemed very odd. I wondered if there was something wrong with those particular folders (maybe a synching or database-corruption issue) in my Yahoo account, and if so, maybe those folder errors were preventing the iPad from downloading and displaying any emails when the account was set up on the iPad as a "Yahoo!" account, but were not preventing the iPad from downloading and displaying emails in unaffected folders when the account was set up on the iPad as an "Other" account.
Final idea (which seems to have worked): I went folder by folder on the iPad mail app (using the Yahoo account I had set up as an "Other" account) and noted which folders would not download and display messages on the iPad. On my desktop, I then used the browser interface to access the Yahoo account. I renamed all the "offending" folders ("[NAME] - OLD"), created new corresponding folders ("[NAME]") and moved all the emails from "[NAME] - OLD" to "[NAME]." After I migrated the email in each offending folder to the corresponding new folder, I re-started the iPad mail app and looked at the new folder in the iPad email app (again, in the account set up as an "Other" account). When i clicked on the name of the new folder (where I had moved the emails), the folder instantly updated and displayed the emails. I continued this process until all the emails had been moved from the offending folders to corresponding new folders. I then deleted all the "[NAME] - OLD" folders. So far, so good. I could now see all the emails in all the folders, but the account still wasn't set up as a "Yahoo!" account so it was missing some of the functionality (such as Reminders). So I simply went back to Settings --> Mail, Contacts, Calendars --> Add Account, again set up the Yahoo email account on the iPad as a "Yahoo!" account, crossed my fingers, held my breath and re-started the iPad mail app. The Yahoo account was back with all the folders. As I clicked on each folder, it immediately re-synched to the Yahoo server and downloaded the contents of the folder. I then deleted the email account I had set up as an "Other" account.
Bottom line: My layman's takeaway from the above experience is that, if any folder in a Yahoo account has some technical problem, the problem apparently prevents the iPad mail app from accessing any folders (including the Inbox) when the account is set up on the iPad as a "Yahoo!" account, but does not prevent the iPad mail app from accessing unaffected folders when the account is set up on the iPad as an "Other" account. If this hypothesis is correct, I found two solutions: (1) set up the account on the iPad as an "Other" account (which at least gives you access to the unaffected folders), or (2) move all the emails from the affected folders to new folders, delete the affected folders and re-install the account on the iPad as a "Yahoo!" account.
I hope this solution works for you, and I hope it continues working for me!