Here's why:
* Internal Evidence: The play references the Lord Chamberlain's Men (Shakespeare's acting company), which was established in 1594.
* Performance Dates: The first known performance was at the wedding of the Earl of Essex in 1595, although this may have been a special, courtly version.
* Quarto Publication: The first quarto (printed edition) of the play was published in 1600, which suggests the play was already popular enough to warrant printing.
So, while we can't pinpoint the exact date, A Midsummer Night's Dream likely belongs to the "middle period" of Shakespeare's career.