Attempt #1 to fix connectivity problems:

After using an old computer for the first time in a while, I noticed it was having the connectivity problem.
So I checked the extension's console and saw that the response from mail.google.com was
"Failed to load resource: the server responded with a status of 401 (OK)"

After manually visiting mail.google.com, the problem went away. This mimic the workaround I tried.

background.js

@@ -12,6 +12,7 @@ var canvasContext = canvas.getContext('2d');
 var pollIntervalDefault = 60;  // 1 minute
 var pollIntervalMax = 3600;  // 1 hour
 var requestTimeout = 1000 * 2;  // 2 seconds
+var tryAgainTime = 1000 * 5;  // 5 seconds
 var rotation = 0;
 var loadingAnimation = new LoadingAnimation();
 
@@ -188,7 +189,7 @@ function startRequest(params) {
   );
 }
 
-function getInboxCount(onSuccess, onError) {
+function getInboxCount(onSuccess, onError, workaroundAttempted) {
   var xhr = new XMLHttpRequest();
   var abortTimerId = window.setTimeout(function() {
     xhr.abort();  // synchronously calls onreadystatechange
@@ -215,6 +216,20 @@ function getInboxCount(onSuccess, onError) {
       if (xhr.readyState != 4)
         return;
 
+      // Check for edge case where mail.google.com has not yet been visited and try again
+      if (xhr.status === 401) {
+        // Visit regular gmail to authorize feed access
+        console.log('Got error 401 while getting inbox count... opening', getGmailUrl());
+        chrome.tabs.create({ url: getGmailUrl() });
+        // Try again
+        if (!workaroundAttempted) {
+          window.setTimeout(function() {
+            getInboxCount(onSuccess, onError, true);
+          }, tryAgainTime);
+        }
+        return;
+      }
+
       if (xhr.responseXML) {
         var xmlDoc = xhr.responseXML;
         var fullCountSet = xmlDoc.evaluate("/gmail:feed/gmail:fullcount",