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Its totally possible that other crawlers can index something, that we are blocked from indexing. I would suggest reaching out to the Support Team with some specific examples so we can see whats going on. help@moz.com Best,

There are a lot of factors that can affect our ability to find and index links to your site. It's important to note that we are always adding new data to our index, but it may take some time for us to discover backlinks to your site based on factors like the crawlability of the referring pages, quality of the links, and the referring pages, and more. If you are not seeing links that you know you have, you may want to make sure that they can be indexed. It is also a good idea to check to see if we've indexed the page on which that link is found. If we haven't indexed the referring page yet, you won't see your link in our index. Lastly, I have a great guide here with some things to check around why we may not have found your links yet: https://moz.com/help/link-explorer/link-building/moz-isnt-finding-your-links