Minutely Log Backups - Missed Jobs
You did not mention your workload, but I run log backups every minute in our OLTP environment without issue (granted, we don't process thousands of transactions per second).  If a log backup happens to take over 1 minute, there is no failure.  The job will merely run again on the minute mark after the current job completes.

As Brent says in What happens to transaction log backups during full backups?


  Backups are beautiful things. You should take full ones, and probably
  differential ones, and if you’re in full recovery model, definitely
  log ones. How often you take them is up to you and your boss. Or maybe
  their boss. But it’s definitely not up to you. Unless you’re your
  boss.
  
  Log backups during full backups won’t hurt anything, and may end up
  helping things if your full backup fails, and you need to restore
  something.
  
  Log backups during full backups will not truncate the log. That has to
  wait until the first log backup after the full finishes.


Some additional information regarding Understanding Schedules and Long-Running Jobs, which I believe still holds true today:


  What happens when a scheduled job runs longer than the scheduled
  interval time period? For example, if a job is scheduled to execute
  every 10 minutes and takes 35 minutes to complete, does the job
  execute again while it's still running?
  
  On the SQL Server Most Valuable Professional (MVP) newsgroup, a
  private forum for SQL Server MVPs, Microsoft published two design
  facts about the SQL Server Agent that can help answer this question.
  First, the SQL Server Agent runs only one instance of a job at a time.
  Second, the SQL Server Agent calculates the Next Run Date at the end
  of the job. So, if a job execution takes longer than its scheduled
  interval, the job will rerun at the next recurrence point. For
  example, imagine that Job A is scheduled with a 1-minute recurrence
  interval. Job A starts at 1:25 p.m. and runs for 1 minute 45 seconds.
  Job A is still running at 1:26, when the next scheduled recurrence
  should happen, and the SQL Server Agent will run only one instance of
  a job at a time—so nothing happens at 1:26 p.m. When Job A ends at
  1:26:45 p.m., the SQL Server Agent resets the Next Run Date. Based on
  the specified 1-minute recurrence interval, the next time the job will
  run is 1:27 p.m.