How can I use Java try without catch?

The classical way to program is with try ... catch. When is it appropriate to use try without catch?

In Python the following appears legal and can make sense:

try:

  #do work
finally:
  #do something unconditional
However, the code didn't catch anything. Similarly one could think in Java it would be as follows:
try {
    //for example try to get a database connection
}
finally {
  //closeConnection(connection)
}

It looks good and suddenly I don't have to worry about exception types, etc. If this is good practice, when is it good practice? Alternatively, what are the reasons why this is not good practice or not legal? (I didn't compile the source. I'm asking about it as it could be a syntax error for Java. I checked that Python surely compiles.)

A related problem I've run into is this: I continue writing the function/method, at the end of which it must return something. However, it may be in a place which should not be reached and must be a return point. So, even if I handle the exceptions above, I'm still returning NULL or an empty string at some point in the code which should not be reached, often the end of the method/function. I've always managed to restructure the code so that it doesn't have to return NULL, since that absolutely appears to look like less than good practice.