I have around 1000 customer to import when I try customer import using magento2 default customer import. Validation works perfectly and when I start import it gives me the message "Import successfully done" but no customer imported.
Make sure you have reindexed all the indexes or your customer grid would not show imported customers. Use following command.
php bin/magento indexer:reindex
Note that if you're only not able to see the customer in admin you can also run;
php bin/magento indexer:reindex customer_grid
this will reindex corresponding to customer grid only and populate data in the corresponding flat table.