Company Names, addresses, dates:
  Fletcher & Co, Harrisonburg VA
  Paul Fletcher's Pharmacy, Harrisonburg VA
  Fletcher Brothers, 2 Court Square, Harrisonburg VA ¹
  Fletcher's Prescription Store, Harrisonburg VA (circa 1940)

Notes:
Isaac Paul Fletcher (1868-1936) was the oldest of 3 brothers.² Apparently he went by his middle name Paul, as that is what is embossed on one of the bottles shown below, and in some of the newspaper references I've seen.
The earliest reference I found to Paul Fletcher's drugstore in Harrisonburg is in 1898³, and the latest one I saw was in 1903⁴.
After that he shows up as a druggist in Richmond from 1910-1916⁵. He ended up in Alexandria VA, where he continued as a druggist until his death in 1936.⁶

Paul Fletcher had 2 brothers, Abner Kilpatrick Fletcher Jr. (1880-1973)⁷ and Samuel Patterson Fletcher (1885-1948)⁸.
I couldn't determine if they simply took over their brother's drug store, or if they started their own drug store sparately, but they were the "Fletcher Brothers". Their drug store shows up in various virginia drug store directories I've found from 1910 to 1938. I don't have access to directories for every year, so they may have started earlier / lasted longer than that.
The book "Images of America Harrisonburg" by Scott Hamilton Suter & Cherly Lyon says on page 26 that Fletcher Brothers occupied the Shacklett building until the 1940s. Page 34 has a photo of the building that replaced the Shacklett building at 2 Court Square.

I couldn't find any references to the name "Fletcher's Prescription Store", but that's what is embossed on one of the bottles below.
I also couldn't find any references to the name "FLETCHER & CO", which is embossed on one of the bottles below. I suspect it was an early endever by Paul Fletcher, because that bottle has a patent date of 1887 on the base, indicating it was likely manufactured within 8 years of that date.

Bottles:

	This is a clear 3" bottle embossed: vertical: PAUL FLETCHER's / PHARMACY / HARRISONBURG, VA.
	This is a clear 3.75" bottle embossed: vertical: FLETCHER & Co. / DRUGGISTS / HARRISONBURG, VA. base: PAT'D DEC 13/87 / C L G CO
	This is a clear 6.5" bottle embossed: vertical: Fletcher's / PRESCRIPTION STORE / HARRISONBURG, VA.
	This is a clear 4.6" bottle embossed: vertical: PURE DRUGS / FLETCHER BROS. / HARRISONBURG, VA. base: C L G CO / 5
	This is a clear 3.3" un-embossed bottle with a label from 1940 that says: label: FLETCHER's / PHARMACY / HARRISONBURG, VA. / ... 5-1-40 ... Dr.Preston

This is a 2.8" tall dose glass. It is marked with a white Applied Color Label: vertical: FLETCHER BROTHERS / PHARMACY / "ON THE SQUARE" / HARRISONBURG, VA. The gradations show up to 4 tablespoons or 1 "wine glass"