The OED defines a tinner as 

1. One who gets or digs tin ore; a tin-miner.

1512 Act 4 Hen. VIII, c. 8 All other tynners..dyggyng of tyn in the severall soyle of the said Richard. 1602 CAREW Cornwall 8b, Where the finding of these affordeth a tempting likelihood, the Tynners goe to worke. 1670 PETTUS Fodinæ Reg. 12 The King for advancement of the Stannaries..frees the Tinners from all pleas of the Natives touching the Court. 1743 WESLEY Jrnl. (1903) 147 Nine or ten miles east of St. Ives, where we found two or three hundred tinners. 1883 R. T. DYER in Leisure Hour Dec. 733/2 In Cornwall, the second Monday before Christmas is a festival kept by the tinners.

2. One who works in tin; a tin-plater, tinman, tinsmith.

1611 COTGR., Estaingnier, a Pewterer, a Tinner. a1817 T. DWIGHT Trav. New Eng., etc. (1821) II. 53 His trade was that of a tinner. 1890 Anthony's Photogr. Bull. III. 45 Have made for you at any tinner's, a tin pan about an inch larger all around than your toning tray.

Tinner also can to be know as an idividual who makes cans or tins.  The can makers predate the 16th century reference.  They may be related to the pewterers and founderers