Can losses in options exceed the premium paid?

For an option buyer, no: the maximum possible loss is exactly the premium paid, since buying an option only gives you the right, not the obligation, to exercise it, and you can simply let it expire worthless if the trade doesn’t work out, without owing anything further. For an option seller (writer), however, the risk is fundamentally different and can be substantially larger than any premium received, since selling a call option carries theoretically unlimited loss potential if the underlying rises sharply, and selling a put option carries a large (though technically capped at the strike price) loss potential if the underlying falls significantly. This asymmetry is one of the most important things to understand before choosing to buy versus sell options: buying caps your downside at the premium, selling does not cap it at all in the case of calls, and caps it only at a potentially very large number in the case of puts.