Trouble connecting to SparkPost - Joomla 2.5.28 / Civicrm 4.6.14
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.