In my adventures with the new web parts I ran into the following error when I added my web part to a page:
Something went wrong
If the problem persists, contact the site administrator and give them the information in Technical Details.***ERROR MESSAGE: ***Failed to load component “4fb1a04a-1d7e-4683-8046-de1646f94fb0” (MyClientSideWebPartWebPart). Original error: ***Failed to load URL ‘https:/ /myclientsidewebpart.azureedge.net/my-client-side-webpart/myclientsidewebpart-myclientsidewebpartstrings_en-us_536e65149b0acf4d52c0043073b9fc59.js’ for resource ‘myClientSideWebPartStrings’ in component ‘4fb1a04a-1d7e-4683-8046-de1646f94fb0’ (MyClientSideWebPartWebPart). There was a network problem. This may be a problem with a HTTPS certificate. Make sure you have the right certificate. ***CALLSTACK: i/<@https://spoprod-a.akamaihd.net/files/sp-client-prod_2017-04-28.005/sppages-assembly_en-us_03284083c09713dfb7fbb719a51b09de.js:144:54695
As always a very helpful error message.
Within my write-manifests.json file and my deploy-azure-storage I used the following two lines:
“cdnBasePath”: “https:/ /myclientsidewebpart.azureedge.net/my-client-side-webpart/”
“container”: “my-client-side-webpart”

Note that the access type is Blob rather than Container!
All I had to do was match everything up and my web part appeared nicely:

So rather than
“container”: “my-client-side-webpart”
I had to go for
“container”: “myclientsidewebpart”