No, you cannot. You can say "me pongo tenso" (I get stressed), "me pongo nervioso" (I get nervous), it is very similar to the use of "get" in front of adjectives. You can use an adjective after "get" and you will make a verb out of the adjective, exampli gratia "the beers are getting cold", "getting" does not have any meaning in that sentence it only makes "cold" a verb. "Ponerse" has the same function in Spanish. "Ponerse de mal humor" (to get into a bad mood), "ponerse de buen humor" (to get into a good mood). "Ponerse contento" (to get happy), "ponerse triste (to get sad). I think you get the idea. Now, the reason why you cannot use "ponerse" as "to become" or "to turn into" is because that would be "llegar a ser" or "convertirse". As in "ella se convitió en profesora" (she became a teacher), "él llegó a ser el mejor de su curso" (he became the best of his class), "Ella llegó a ser presidente" (she got to be the president). If you need more examples of further explanation, please, feel free to ask me.