The short answer is: yes. The long answer is: depends upon the country and their passport regulations.
In the case of dual citizenship, it is totally valid for a person to hold two passports. Countries like Australia, Switzerland, France, Greece and most European countries allow dual citizenships. Two passports are also a regular occurrence for people who are working for the governments, such as diplomats.
Dual citizenship can be obtained in cases when a child is born in one country, but its parents are citizens of another. In some circumstances and if the countries’ laws allow that, a child can hold two citizenships inherited from its parents.
Citizenships and passports can also be gained through marriage.
Of course, naturalised citizens are also eligible for citizenships. Every country has its own regulations when it comes to this, like the number of years that need to be spent in the country, knowledge of language and so on. Some naturalised citizens can keep the citizenship of their home country, but others are not allowed.
There are, however, possible scenarios when it is legal to have two valid passports without a dual citizenship. In the U.S. for example, if a citizen applied for a visa and had to send their passport to the embassy, but suddenly they need to travel to another country, they can be granted a second passport.
Also, it is possible to get this second passport if a passenger has a proof of planned travelling to two countries that don’t see eye to eye. It is common that the border officials of one country won’t let you in if they spot a stamp from the opposite country. In cases like these, a person can apply for a second U.S. passport. This second passport is, however, only valid for two years. Yet it can be renewed. The standard U.S. passport is valid for 10 years.
This similar passport system is also present in the UK. The UK’s second passport is called a “concurrent” passport. They are used by people who are frequently travelling for business purposes and thus applying for multiple visas.
UK citizens can also apply for an Irish citizenship, thus hold an Irish passport. This is possible even in cases when a person has only one grandparent that is Irish.
A bit pricier second passport can be gained if you buy property or invest in other countries and get their citizenship.