This is a fun question that comes up often. When it comes to pre lamped light strings (5mm/m5/C6/G12 and even the non removable bulb C7/C9 strings), the simple answer is yes, if one bulb burns out, the rest will stay lit. The more complex (and relevant) answer is that LED strings don't work the same way that incandescent strings do. Hence, the likelihood of a bulb burning out is pretty slim. With LED strings, when there are issues it is almost always relative to the inline rectifier having a problem such as inconsistent electrical supply (ie dirty power, bad voltage drops/spikes, etc) or critters chewing through wires. The good news is that rectifier failure is covered under our 3 season warranty.
For pro grade strings with screw in LED bulbs, ie C7/C9 stringers with screw in bulbs - if a bulb goes out, the rest will always stay lit as long as the string is in tact.