No, the assignee is a single user field, it always contains one single unique user (or unassigned if you have that enabled)
I know assigning to multiple users has uses, as does assign-to-group, but Atlassian do not implement these.
You can try using a multi-user-picker field instead, but it won't have the functionality of assignee.
Edit: See https://jira.atlassian.com/browse/JRA-1397 for the requests to fix this problem, the reasons Atlassian have chosen not to fix it, (which are wrong IMHO, although they do suggest better ways to do it) and plenty of workarounds.
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