That relief should be temporary. Why? Because the poll numbers serve to reinforce the decidedly uphill climb Trump faces to 270 electoral votes this fall.

Let's game out a few scenarios using our handy-dandy electoral map tool.

Start with the best-case scenario map for Trump and Republicans in which he holds every state Mitt Romney won in 2012 and adds Ohio, Florida and Pennsylvania to the GOP column.

Trump wins 273 electoral votes -- and the White House -- under that map. But his margin for error is tiny -- three electoral votes! -- and this map assumes that he not only wins Florida and Ohio, both of which President Obama carried in 2008 and 2012, but also Pennsylvania, which no Republican presidential candidate has claimed since 1988.

Now, let's assume a Trump victory in the upper Midwest -- so give him Pennsylvania and Ohio -- and a loss in Florida due to the state's large Hispanic population. Here's what that looks like:

This map produces 294 electoral votes for Clinton and 244 for Trump.

Then there's this map, which gives Trump Ohio and Florida but has Clinton winning Pennsylvania:

Trump loses that one too; Clinton ends up with 285 electoral votes to 253 for Trump. So: Even if Trump wins every state Romney won in 2012 and the two biggest, swingiest states in the country, he is still 17 electoral votes short of what he needs to be president.

These polls are good news for Trump as they suggest a level of competitiveness with Clinton in swing states that many Republicans wondered whether he could muster. But even if Trump is able to continue his competitiveness in these states, it will take something close to a running of the table for him to get to 270.