Let me tell you how it works.
Say you have an Adele's track. In order to display Adele's picture in the thumbnail browser, MB would look at "Thumb" sub-folder of artist picture folder set in Preferences> Tags (1) settings. That's the only place it's looking at for artist picture and won't look at any embedded, linked pictures for that. And if it can't find Adele.jpg (or png) file in that folder, it retrieves one from the Web. If the downloaded image is in a good quality, then it gets saved as Adele.jpg in Thumb folder, and MB will always use that picture for Adele without downloading a new one. If the downloaded picture is not good (same or small than 128x128px), then it remains only as image cache without getting saved in Thumb folder. Next time it will repeat the process. So if you notice MB keeps downloading an artist's image, that means MB has not found a good quality picture for the artist from MB's image providers.
Then how can you make it show your custom image for Adele? Add the file as Adele.jpg to Thumb folder. But that would be pretty cumbersome. An easier way would be copy the image (adding to the clipboard) and then paste to Adele's artist picture in thumbnail browser using "Paste artwork" context menu, then MB saves the picture to the Thumb folder with proper naming.
If you have artist pictures embedded to files, you can use track information panel to copy the picture. Right click on the artwork of the track info panel, select the embedded artist picture, and then select "Copy" context menu. Then just paste it to the thumbnail browser.