The numbers of days and month in a date are ordinal numbers. But the year is a cardinal number. The suffixes -te and -ste can only be used for ordinal numbers:
Der zehnte Versuch.
The tenth try.
Der siebenundfünfzigste Geburtstag.
The fiftyseventh birthday.
Der vierte Juli.
Fourth of July.
Ich wurde am neunzehnten sechsten Neunzehnhundertfünfundsechszig geboren. (It's dative case here, you have to an extra -n at the end.)
Literally: I was born at the nineteenth sixth nineteenhundredsixtyfive.
In »Die 1960er Jahre« you use 1960 similar to this cases:
Die Wiener Kaffeehäuser
The Viennese Coffeeshops
Die Londoner Taxis
The London Taxis
Die Habsburger Monarchie
The Habsburg Monarchy
Die 1960er Jahre
The words Wiener, Londoner, Habsburger from the examples above belong to a special class of adjectives, the »non flectional adjectives«, maybe also called »undeclinable adjectives«. The German name is »nicht flektierbare Adjektive«.
This is an example for normal adjectives which can (and have to) be declined:
Ein altes Haus verfällt. - An old house decayes.
Das alte Haus verfällt. - The old house decayes.
Die alten Häuser verfallen. - The old houses decay.
In each sentence the subject is the first three words, the last word is the predicate. The subject is always in nominative case, and here it is a nominal phrase, wich consists of an article, an adjective and a noun.
The first two sentences are in singular, the third is in plural. The first one is indefinite, the last two are definite. This gives, here in my examples, three combinations of number and determination. All three parts of the subject must be declined to this combination, and I have chosen the examples such, that we get three different versions of the adjective (altes, alte, alten). This is the normal behavior of adjectives.
There are three ways to use adjectives:
- Adjective used as attribute: »Ein schnelles Auto« (A fast car)
- Adjective used as compliment of the predicate: »Das Auto ist schnell.« (The car is fast.)
- Adverbial used adjective: »Georg fährt schnell.« (George drives fast.)
But there are adjectives, that can not be used in this way:
- Ein Londoner Taxi. - A Londoner taxi. - OK.
- Das Taxi ist Londoner. - The taxi is Londoner. - No, this is nonsense!
- Georg fährt Londoner. - Georg drives Londoner. - No! Bullshit again!
If an adjective only can used as an attribute, then in German it will not be decliend:
Ein Londoner Haus verfällt.
Das Londoner Haus verfällt.
Die Londoner Häuser verfallen.
Adjectives, which behave this way, are called »nicht flektierbare Adjektive«. A subset of this set are the adjectives, that are derived form proper names which end in -er, like in the examples above. Those adjectives have an additional exception. They not only can't be declined, the also have to be written with a uppercase first letter:
Ein Londoner Taxi.
Ein britisches Taxi.
When used in a construction like »Die 1960er Jahre«, then the year (1960) behaves exactly like one of those proper names that need an -er-suffix when used as an adjective. This means: It becomes an adjective, that can not be declined.