Those who say that there is "definitely no apostrophe" in date years, such as 1990's, are definitely wrong. Likewise, those who say that an apostrophe is wrong in the plural of abbreviations, such as MBA's, are also wrong.
The rules about the use of apostrophes to signal a plural rather than a possessive may seem complex, but for the purpose of the current question, can be reduced to saying that an apostrophe is used to signal a plural for a word that does not otherwise have a "natural" English plural. Therefore, in strictly correct terms, an apostrophe is inserted in year dates, abbreviations, foreign words that would be rendered in italics, and certain English words.
As always, examples are the best illustrations of the rules.
The plural of dog is dogs, because dog is a word with a "natural" English plural. Therefore, we can say "It's raining cats and dogs". But "if" and "but" are English words with no natural plurals, hence we would say something like "no if's or but's, do your homework!" That is, an apostrophe is used to signal the plural. Likewise, individual letters of the alphabet have no natural plural, so we would say something like "How many a's are there in supercalifragilisticexpialidocious?"
Foreign words follow the same convention, particularly where the word or words would be rendered by a printer in italics, to show the foreign origin. Thus "lingua franca's". This convention however is not usually applied to foreign words that have become so integrated into English that they are treated as native words.
As will readily now be evident, abbreviations and year numbers likewise follow this convention.
That said, the convention for years and abbreviations is falling out of use, and 1990s and MBAs are increasingly seen in contexts where it is not ignorance of the convention that has resulted in dropping the apostrophe, but a conscious editorial decision.
However, the original question asked if "it is ever right" to use the apostrophe to signal plurals in years and abbreviations, and the answer is decidedly yes.
Incidentally, the custom in the US, which tends to be more rigorously correct in these matters than British English (surprisingly to many), is to retain the apostrophe in these special plurals.