I installed the SparkPost/CiviCRM extension and followed the README and all seems correct but I can not connect to SparkPost SMTP server. Is anyone else seeing a problem?
One possibility is that you are trying to send an email with a "From address" that is not in the sending domain(s) that you have verified in SparkPost. I think they only allow email from verified domains, so you can't send "from" a Gmail address in CiviCRM, for example.
If you are having trouble with the PESC Sparkpost extension you may want to try the one from Cividesk. It only requires inputting the API code and is pretty much plug and play.
I don't know about SparkPost specifically... however I use similar services with my CiviCRM installs...
Error seems to imply that your username and/or password is wrong: https://support.sparkpost.com/customer/portal/questions/16067096-atomic-email-sender-errors?b_id=7411
added in response to first comment
I ran into that on Google Compute Engine hosting. All mail ports but the non-standard 2525 were blocked. https://support.sparkpost.com/customer/portal/articles/1988470-smtp-connection-problems
I would test the ports. ssh into the server and telnet the ports to see if you can talk to the mail server. You should get a response something like below (try 587 and 2525):
david@server ~ $ telnet smtp.sparkpostmail.com 587 Trying 126.96.36.199... Connected to smtp.sparkpostmail.com. Escape character is '^]'. 220 2.0.0 smtp.sparkpostmail.com ESMTP ecelerity 188.8.131.52068