I couldn't get my new Echo to connect and Amazon's support staff got it working. The issue was actually with my Verizon router. The Echo uses some specific ports to connect to the router and when mine was installed, the installer enabled the router firewall and set it to maximum security for both frequencies. This, of course, blocks all traffic on all but the basic ports needed for internet traffic. The Echo ports were being blocked. The firewall was reset to "Typical Security" and the echo connected right away.
The story:
I got an Echo for Christmas and, just like seemingly millions of others, it wouldn't connect to my WiFi. I spent a week tryig everything I could find via Google: unplugged the Echo for three seconds, reset the Echo, uninstalled and reinstalled the Alexa app, rebooted my router and reset the Echo, attempted to connect to both 2.4 and 5 GHZ networks, created a guest network and tried to connect to that, ensured the 2,4 and 5 GHZ networks have different SSIDs, and tried different channels on both by using a WiFi analyzer app. to determine the best signal strength. I finally called the Echo support number (1-877-375-9365. Great support, by the way). The tier 1 tech spent a few minutes going through the checklist of things I had already tried and didn't even force me to go through the whole thing again, but only the basics. He then transferred me to the next level, a cheerful young lady who spent over two hours on the phone with me, exhausting all possibilities. She determined that the problem was with the router settings and tried to help me find them. Between my ignorance and the difference in terminology that different routers use, she suggested she call Verizon for help. While I was on hold, she got in touch with Verizon support and created a conference call between the three of us. She had the Verizon rep. check a few settings and asked her to check some UDP ports. After the Verizon tier 1 tech ran out of knowledge, she patched us to network support. The gentleman that answered was very helpful and obviously experienced. The Amazon and Verizon reps worked together and quickly reached an understanding. The Verizon IT rep remoted into my computer. The Amazon rep. gave him the MAC address of my Echo and the Verizon rep. discovered that the Echo was actually connected to the router. The Amazon rep. gave him the list of UDP ports the Echo requires to be open and he added them via port forwarding. The Amazon rep. then asked him if the firewall was active. When the Verizon rep. looked at the firewall page, he noticed the firewall was not only active, but was set to "Maximum Security" and explained to me that it was blocking all outgoing and incoming traffic, other than the ports needed for basic internet functions. The Amazon rep, asked him to disable the firewall, which he did and asked me to go through the Echo setup with the Alexa app. again. This time, the Echo connected without issue. As a test, she told me to ask Alexa the time. Eureka! It answered with the correct time.
Sorry for the extremely long post, but I hope that this may help others. Sometimes the answer can be found, not in the direct cause of my problem, but in the details of the issue and the steps taken to correct it.