Jira's attachment functions are very simple, and it's not really a "field". The "field" you see is simply a placeholder that lets you talk to the list of attachments for an issue.
Even if you could add more than one "field", there's no functionality for holding attachments differently or separately, they all go in the same place.
I am not sure I understand your question correctly. If you want to attach multiple files to an issue, you only need 1 attachment field to perform this. When following the link, you should be able selecting multiple files from within the same folder (hold Ctrl and click all files you want to attach) or navigate to the field more than once in case you require to attach files from multiple location.
If you want to differentiate screens between issue types, I would advice you to have a look at this and subsequent documentation regarding fields, screen and the schemes related to them.
My original answer still stands - attachments on an issue are a flat list.
The attachments "field" does not hold any data, it's a placeholder for you to access the list of attachments.
So you can't have 4 fields, and even if you did, they'd all do exactly the same thing and land in the same simple list of attachments.
"Smart Attachments" is yet another commercial JIRA plugin.
From a quick glance, the plugin promises some interoperability with MS Office, but the description doesn't say "use multiple attachment fields on the same issue" so I doubt it'll do more than modify the Atlassian attachment thing.
So the answer that Nic Brough has given four times in this thread seems to be absolutely correct: You can only have one attachment field per issue.
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