Yes, if you trade on leverage, then the face value of your trading position is more than the money you have in your trading account. Therefore, if the market moves against your trade enough, it could take your account balance negative.
For example, suppose you open a position to trade one standard lot of USD/JPY. That's 100,000 units (100K) of base currency which in this case means 100,000 US dollars.
If you have $25,000 in your account, then your effective leverage would be 4:1, since your trade has a face value four times greater than your account balance. With this leverage, it would take a 25% move against your trade to take your account balance to zero. Therefore, a move of more than 25% (about 2,500 pips) against your trade could leave you with a negative account balance.
If you have $10,000 in your account, then your effective leverage would be 10:1, so a move of more than 10% (about 1,000 pips) against your trade could take your account balance negative.
At the time of this post, the US regulators, have set the minimum margin requirement for USD/JPY trades at 4% which allows you up to 25:1 leverage. That means you would need at least $4,000 in your account to open a 100K USD/JPY position. Trading with this maximum leverage would mean a move of more than 4% (about 400 pips) against your trade would leave you with a negative account balance.
It's important to note that under normal market conditions, most brokers including FOREX.com have monitoring systems in place that will automatically close your trade once your account falls below the minimum margin requirement. This is to reduce the chances that your account balance goes negative.
That said, you do risk incurring losses greater than your account balance, especially during periods of extreme market volatility. While it is not FOREX.com’s policy to hold clients responsible for modest negative balances, we do reserve the right to hold clients responsible for large debit balances and when special circumstances apply.
There are cases where the market price gaps or jumps from one price to a drastically different price in a short space of time sometimes even from one tick to the next. This can happen over the weekend with a gap between Friday's closing price and Sunday's opening price.
It can also happen with a surprise economic announcement, such as occurred on January 15, 2015 when the Swiss National Bank said they would no longer support the 1.2000 floor on the EUR/CHF exchange rate which had been in place for years. You can read more about what happened then in this post: https://forums.babypips.com/t/low-leverage-is-in-fact-dangerous/112433/3
It's worth noting that after the SNB event, some brokers including FOREX.com forgave negative balances, while some other brokers did not. That said, no broker in the US can guarantee that you will not be held responsible for a negative balance on your account resulting from trading losses. Therefore, it is important for you to use leverage responsibly, just as you should seek out brokers who demonstrate that they offer leverage responsibly.
FOREX.com and our parent company GAIN Capital believe that how we raised margin requirements ahead of the SNB announcement demonstrates our responsible approach to leverage. We pride ourselves in taking a long term view in the best interests of our customers, our employees, our business partners and our shareholders. It was by no means a popular choice at the time when we raised margin requirements on Swiss Franc pairs, but we think history has proven it to have been a wise choice for all our stakeholders.