Create and back up files
In my opinion your code it's not so bulky but I would do something like below with few changes. I have not tested it too much but it should works.

I will use FilenameUtils for easy manipulation with files like getting basename or extension of file - it will save you some of the lines and also it is one of the most used library for file manipulation. 

I would remove String getFileName and String getFileExtention and create self-described function and delete in code comments. I also removed return false for InterruptedException - I would like to get another draft of the name in the next iteration.

I dont like spaggetti code when some string is creating like (with +): 

return name + dateFormat.format(new Date()) + ext;

so here I would use StringBuilder or String.format or something like that.

Below is what I have done in few minutes.

import org.apache.commons.io.FilenameUtils;

import java.io.File;
import java.io.IOException;
import java.text.DateFormat;
import java.text.SimpleDateFormat;
import java.util.Date;

public class FileBackup {

    private static final String pattern = "-dd_M_yyyy-hh_mm_ss.";
    private static final DateFormat dateFormat = new SimpleDateFormat(pattern);

    /**
     * Creates new file. If file with given name exists, then backups
     * it by renaming and adding time stamp to file name;
     *
     * @param filePath the fully qualified name of file

     * @return the file on success; otherwise returns null
     *
     * @throws IOException
     */
    public static File backupAndCreateFile(String filePath) throws IOException {
        return failedBackuOfExistingFile(filePath) ? null : new File(filePath);
    }

    private static boolean failedBackuOfExistingFile(String filePath) {
        File fileOnPath = new File(filePath);
        return  fileOnPath.exists() &amp;&amp; !successBackupFile(fileOnPath);
    }

    private static boolean successBackupFile(File fileToBackup) {

        while (true) {

            File backupDraft = new File(generateBackupName(fileToBackup.getAbsolutePath()));
            if (!backupDraft.exists()) {
                return fileToBackup.renameTo(backupDraft);
            }
            fileWithGeneratedNameExistsNowSleepForSomeTime();
        }
    }

    private static void fileWithGeneratedNameExistsNowSleepForSomeTime(){
        try {
            Thread.sleep(500);
        } catch (InterruptedException e) {
            e.printStackTrace();
        }
    }

    private static String generateBackupName (String filePath) {
        String fileName = FilenameUtils.getBaseName(filePath);
        String fileExtension = FilenameUtils.getExtension(filePath);
        return new StringBuilder().append(fileName)
                .append(dateFormat.format(new Date()))
                .append(fileExtension)
                .toString();
    }
}