The summary of this Suggestion does not seem to match its description and comment thread and I assume this was a typo when originally created.
I seems that
As a scrum master I would like to be able to have one sprint active per rapid board even if they are in the same project
was meant to be
As a scrum master I would like to be able to have more than one sprint active per rapid board even if they are in the same project

Note that this was recently shortened by myself from
As a scrum master I would like to be able to have one sprint active per rapid board even if they are in the same project
to
Ability to have one sprint active per Agile board even if they are in the same project
which does not change the relevant part of the summary.

Given this, it would seem that the Parallel Sprint feature is all that is required to satisfy the requests here.

The Parallel Sprints feature has now moved from JIRA Software Labs to a configuration option in JIRA Software Cloud (since 11 Jan 2016), please see https://confluence.atlassian.com/display/JIRASOFTWARECLOUD/Using+Parallel+Sprints

This change will also be shipped in the next 7.x version of JIRA Software Server. For further progress please follow https://jira.atlassian.com/browse/GHS-11123

A related request for Parallel Sprints to be a board setting rather than a global setting can be tracked at https://jira.atlassian.com/browse/GHS-12588

Kind regards,
Martin
JIRA Software

Our teams just ran into the same issue, although we want separate teams to have separate boards with separate filters that aren't necessarily fully exclusive. It's crazy that having multiple separate teams that share one project/component is so difficult to manage, especially since other tools I've used do this just fine.

https://answers.atlassian.com/questions/33142331/cant-start-my-sprint-because-of-another-teams-active-sprint

I see this issue is an old one. Are there plans or not to implement this?

Thanks.

Hi Atlassian,
Is there any plan to develop this feature ? This is causing a lot of issues for our dev teams here.

Vote up for this feature.

This is really needed for matrix organizations

My approach to this with 6.2 is to have a select custom field named Team and to create Swimlanes based on Team=TeamA, Team=TeamB etc. Card coloring by Team also helps plus Quick Filters. That gives me multiple ways to focus on a single team in a board. Still this is for a single sprint.

Just adding a note here for if/when this gets picked up for future use with a couple of our use cases for overlapping sprints. In our world, every code change is tied to a JIRA card and every release is tied to a sprint. If it doesn't have a sprint, it doesn't have a release.

Use Case 1: Sprint "done", regression uncovers bug that needs to be fixed
-Scenario
--Sprint 1 ends on Thursday 1/15 and releases on Tuesday 1/20
--Sprint 2 starts on Friday 1/16
--Sprint 1 is "complete" but on Monday 1/19 we uncover an issue that needs to be resolved (ISS-10) before golive. This occasionally happens with regression testing or something slipped through. We leave Sprint 1 open so that a developer can go in an make the fix via an issue in Sprint 1. They won't get credit for it in the burndown of Sprint 1 or Sprint 2 (which I don't have a problem with) but if we are forced to close Sprint 1 before we start Sprint 2 things become a mess. If ISS-10 is created as a card in Sprint 2, it is not really true because the code change will go out with Sprint 1.

Use Case 2: Hotfix
-Scenario we are in the midst of Sprint 20 and an unscheduled release is identified
-Let's say we just released Sprint 19 and are in the midst of Sprint 20 and an issue crops up that requires a deploy. What we do on the classic boards is go in and create Sprint 19.1 that will contain the issues for the hot fix. This minisprint may only live for one day but all of the issues for the release will have a sprint.

In both of the above examples, the sprint teams won't get credit for the cards in their current sprint (which I am fine with because this averages out over time and is why velocity is never 100% of available time) and we know that.

The Parallel Sprints feature seems like it should allow everything to work correctly for us but when that feature is disabled, I wanted to provide some use cases to help in crafting a longer term solution.

In general many companies are working in larger environments with multiple teams, sometimes with independent schedules. In our case the hardware developers and the software developers. They are working towards the same milestones but in between the schedule for a "device driver" (SW side) is independent of the schedule for a "Lab prototype board" (HW side). Therefore I have added another feature request GHS-7203 which is loosely related to this one.

Thanks, sclowes. I somehow missed your post, so thanks for pointing it out. This is exactly what I was looking for.