You are suffering from a really simple problem, but a common one for those new to the language. It is how you are reading in your data from the user. First you use
console.nextInt() which has one side effect when collecting user input, it leaves carriage returns in the buffer. It just reads in the integer. Right now it isn't an issue, but when we do the next change you will see why it will be...
The second part is how you read in the student's name. Using just
next() of the scanner object will read tokens... in this case tokens separated by a space. So I was to enter "Martyr Two" the first call to next() would read in "Martyr" and the next call would read in "Two". To fix this, use another method like
nextLine() which will read up to the first line break, thus read in the entire students name.
Now why this little nextLine() change will be an issue with your first one is because when the user enters a value like "2" for the number of students and hits enter it will read in the 2 and leave the carriage return for your nextLine to read, which will make it appear to skip over your first student and repeat the prompt for a student's name.
So how do we fix it? Glad you asked. The trick is to go ahead and make the change for the student's_name to use nextLine() and back for collecting the number, use nextLine as well, but parse it into an integer like so...
CODE
// Collect what they typed, then change it to an integer.
class_limit = Integer.parseInt(console.nextLine());
The beauty of this is that it will read up the carriage return as well and leave nothing behind, then convert what they typed to a number for use in the rest of your program. Then your other nextLine() calls will work as expected and collect their full name.
One warning though, when collecting the number in the class, make sure you check if the user entered a number. If they enter letters or characters, it will throw an error in your parseInt (which you could trap if you like).
But since it is a simple program, you can choose to let it go and assume the user always enters a number.
Hope this helps your understanding. Let your instructor know what you have learned here.

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