Usually, no. The broker will not let you lose more than you have in your trading account. It will simply close your losing position if the account balance approaches zero. If you lost more than you invested, it would be a direct loss to the broker. It is in the interests of the broker to prevent such losses. To protect themselves, brokers introduce the concept of a stop-out level (usually around 20%). This means that the most unprofitable position will be closed as soon as (Equity / Used Margin) × 100% is equal to or less than this level.
In rare cases, slippage and a significant price gap can lead to a negative account balance. However, brokers rarely go after traders to recover negative balances.