According to an article in The Upland Almanac Linsley Brothers gun firm has been in business for 225 years. It was actually founded by William Calvert of Leeds in 1780, but George Linsley acquired the firm in the 19th century. The article goes on the say that like other Birmingham gunmakers Linsley bought from the trades, "but only from the highest skilled and accomplished free tradesmen and their guns reflect the fact.
Control passed from George Linsley to Steward Mackie and upon Stewards death in 1937 his daughter took over and she already had 20 years in the business.
They made many fine doubles throughout their history and are perhaps best known for their 12 bore boxlock 2" double game guns. The craftsman shaped and matched a 20 gauge action to a specially built 12 guage steel barrel and the lightest walnut they could find. The end result was a 5 lb to 5 lb 5 oz gun that shot 3/4 oz or 7/8 oz loads. Popularity for the gun faded after WW II. A gun like this might bring $5,800 to $6,200.
This does not tell you much about a 16 ga, but the article does talk the maker up. As so many say you really need to buy the gun, but it sounds like they were a highly respected maker.