Status: Unsolved. Maybe Shakespeare?
This is a tricky one, as it actually does come from The Passionate Pilgrim, a known Shakespeare source (albeit a less commonly cited one). So at first glance you could easily make the case that this is a Shakespeare quote.
Here’s the problem — that collection is actually a bunch of smaller writings from a bunch of authors, and only a few of them have been identified as Shakespeare:
First published by William Jaggard in 1599, this collection of poems, in its entirety, is commonly attributed to Shakespeare. However, a number of the poems were written by others including Richard Barnfield, Bartholemew Griffin, Christopher Marlowe, and Sir Walter Raleigh. Shakespeare has been identified as the author of five poems: numbers I, II, III, V, and XVII.
The quote in question comes from poem #21. Strangely enough, when googling for details on this particular poem I actually landed on myself (via ShakespeareGeek.com).