When considering the overall strength of the Union army tables like this are usually brought up:
This table, from the OR (OR 3, Vol 2, 957) shows an aggregate strength of the Union Army as 664,163. This is quite a force. Compare with the number of regiments published a month earlier:
Now, lets consider the average strength of a volunteer infantry regiment (counting an independent company as 1/10th of a regiment). There are 816.7 Regiments with an aggregate PFD of 573,420 and an aggregate absent, sick or wounded of 107,109. That means the average Union regiment just before the Battle of Fredericksburg was 702 PFD and 131 absent, sick or wounded (833 aggregate present and absent).
Doing the same for the cavalry yields 89.7 regiments with 700 PFD and 133 absent, sick and wounded (833 aggregate present and absent). The artillery has 28 regiments (of 12 companies) with 958 PFD and 176 absent, sick or wounded (1,134 per regiment).
Now, have you ever read of 700 man regiments in the field? They're incredibly rare.
Burnside's Army of the Potomac at Fredericksburg has 265.5 infantry regiments (including a few regular units not in the volunteer count above), in round figures a third of the Union infantry. They also have 18.9 cavalry regiments and 70 companies of artillery. If they were "average" as above they'd have:
186,231 infantry PFD
13,230 cavalry PFD
5,588 artillery PFD
= 205,049 men PFD
In fact (using his 10th December 1862 return (ref))his three active "Grand Divisions" used in the count above only sum to 118,952 PFD and only 103,190 infantry PFDE (adding in the Vol engr bde it becomes 104,695). Using the latter figure the average infantry regiment in Burnside's army (accounting for ca. a third of the total infantry) is 394 (56% of the count above). The PFDE (all arms) of the three Grand Divisions is 114,612 and the aggregate present 138,927, or 17.5% of the aggregate present are not PFDE, this is the sick list (at ca. 1 in 6 is absolutely typical of an army in the field). The PFD or PFDE must be reduced by taking out the daily and special duty men to arrive at an effective combatant strength. We can estimate the Grand Divisions had 90,000 with the combat units and around 77,000 effective infantry (about 71,000 enlisted effectives, the normal CS measure). Thus the musket strength of an average infantry regiment in Burnside's army is now down to around 266 muskets.
If this was typical of the entire Union Army then not 220,000 muskets would be in the field. It is this thinking that has me doubting that the Union ever had 600,000 effectives in the field. I tend to regard Union strength as around 300,000, maybe 350,000 depending on exactly when we measure.