Identity Column in Concurrency, Multithreading, Parallel Processing

Our company would like to understand the effects: - If we have single table with Primary Key Identity column, int primary key IDENTITY(1,1), - and the Application is conducting multi-threading parallel processing, trying to conducts 1000s of simultaneous inserts on one table. Would the identity primary key column fail, with a primary key violation in any way, for example: processors are trying to input the same identity number? Or was SQL designed to be successful even in multithreading parallel insert? Note: IDENTITY_INSERT is OFF