SMTP connect() failed - WordPress

I'm non technical so please bear with me.
We have a WordPress website that has a number of plugins which send out e-mails for example, a form plugin, security plugins etc.
I have an issue where I found that the website wasn't sending any messages to the intended recipients.
Our website is hosted at one company and the e-mails reside at another.

I've contacted both the hosting company and the e-mail service provider and they stated that all is well at both their ends but I'm still not able to send messages from the website.

This is an example of the error message:
SMTP connect() failed.

I suspect that this is an issue at the Hosting end but after speaking with one of their reps again yesterday, they seemed more interested in telling me what they won't do rather than what they would.
They ran a script which sent a message from the hosting server but this was from their server not the SMTP server.
They also stated that they did not have any blocks in place to prevent the SMTP connection.

Does anyone have any idea what I can do to move things forward?

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Marcus BointonCommented:
Have you tried the tests described in the troubleshooting guide? Don't believe your ISP when they say they're not blocking SMTP - check it yourself!

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canuWebmasterAuthor Commented:
Unfortunately Marcus, the troubleshooting guide reads like Mandarin to me. I'm not a programmer and don't know any PHP.
From the very limited description of the problem, could the ISP be blocking the SMTP connection?
Marcus BointonCommented:
Yes. It's very common. Alternatively, they could be intercepting SMTP traffic and redirecting it to their own server, which may also prevent things working as you expect - for example it will cause TLS connections to fail.
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canuWebmasterAuthor Commented:
OK, thanks Marcus.
Is there a more simple to understand way of diagnosing this issue?
Thanks for your time.

Marcus BointonCommented:
Not really. It's a technical problem with technical solutions. Without being able to test it properly, we are working blind.
canuWebmasterAuthor Commented:
I used another plugin to try and diagnose the problem and found that the SMTP ports were being blocked by the mail ISP which I will address with them.
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